Mathematical games interesting to both you and a 5+-year-old child

Question

Background: My daughter is 6 years old now, once I wanted to think on some math (about some Young diagrams), but she wanted to play with me... How to make both of us to do what they want ? I guess for everybody who has children, that question comes up. Okay, I said to her: let's play a game which I called "Young diagram" for her: we took a sheet of paper and I tried to explain to her what a Young diagram is, she was asked to draw all the diagrams of some size n=1,2,3,4,5...

Question: Do you have some experience/proposals of "games" which you can play with your children, which would be on the one hand would make some fun for them, on the other would somehow develop their logical/thinking/mathematical skills, and on the other hand would be of at least some interest for adult mathematicians ?

Related MO questions:

“Mathematics talk” for five year olds it is quite related to the present question, but slightly different - it is about a single presentation to children, while the present question is about your own children with whom you play everyday, you can slightly "push", and so on...

How do you approach your child's math education? it is also related, but the present questions has a slightly different focus: games interesting for children and adults. The book by Alexandre Zvonkine, "Math for little ones" (in Russian here), recommended in answer there - is really something related to the present question.

Which popular games are the most mathematical? is NOT directly related, but may serve as kind of inspiration for answers...

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I think Allen Knutson's answer on “Mathematics talk” for five year olds:

I've spoken (to 5+ years old) about the "puzzles" that Terry Tao and I developed for Schubert calculus, like the left two here:

can be a nice example of an answer to the present question as well: on the one hand there is something to explain to the child and some colorful pictures, and on the other hand that is about research level math ...

Answer 1

One evening at the dinner table, when my oldest daughter was 3 or 4, I was in a teasing mood, and I called her a goose. She didn't want to be a goose, so she refuted the claim, "I am not a goose!" Then I told her to prove me wrong. After some back and forth, she realized that her cause would benefit from some distinguishing feature: "A goose has feathers, but I don't have feathers, so I'm not a goose." I was impressed, so I chose not to continue the teasing by concluding she was a plucked goose.

So began our game "Prove me wrong," in which I make wild claims for her to refute. In the modern version of the game, I will respond to her "proofs" with more refined claims. As a mathematician, it is quite the guilty pleasure to construct these logically sound but apparently absurd refinements. For the child, the game presents a fun way to navigate silly ideas. In the end, she's refining her ability to apply basic logic.

On a good day, I will bring "Prove me wrong" into the classroom. When I introduce matrix multiplication in linear algebra, everyone has seen it before, and so I inject some "fun" by claiming that multiplication is commutative. The more outspoken students read my smile and speak up with an emphatic "No, it isn't!" I then proceed to make my case by multiplying $1\times 1$ matrices and $2\times 2$ matrices that happen to commute. Eventually, a student suggests that I put variables in the entries of my $2\times 2$ matrices.

Answer 2

The game "Set" seems to fit the bill. It's a card came where there are cards that show images which have four different features, each of which comes in three possibilities:

• number (1, 2, or 3 objects)
• color (green, blue, pink)
• shape (diamonds, rounded rectangles, "tildes")
• filling (empty, filled, half-filled)

so there are $3^4 = 81$ cards. You lay a certain number of card open on the table and the players need to find "sets" of cards, and a set are three cards such that on these three card each feature is either the same or all three versions appear. So, this picture shows a set:

In mathematical terms, you are looking for lines in four-dimensional space over three elements.

Granted, it's not easy for 5year olds, but I've met some kids at that age who could play it and had fun.

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One successful way to play it with kids even as young as 4 is to first find the set yourself, and then hand two of the cards of it to the kid. And let them find the third card. You coach them along: "What color is this? What color is this (second card)? So, what color will the third card have to be?"

If it's too hard for them, let them play with a reduced deck for a while: use all the solid cards only, to make a deck of 27 cards, and play with that. Then all the single-shape cards (again 27), so they get used to spotting differences of shading.

If you're going to be playing with younger children for a while, you could consider getting Set Junior. It only includes the solid cards, and the cards are thicker cardboard tiles. It also includes an easier variation, where one is just trying to match the cards in one's hand to existing Sets on a game board.

Answer 3

Another topological game: Sprouts.

Rules:

The game is played by two players, starting with a few spots drawn on a sheet of paper. Players take turns, where each turn consists of drawing a line between two spots (or from a spot to itself) and adding a new spot somewhere along the line. The players are constrained by the following rules.

* The line may be straight or curved, but must not touch or cross itself or any other line.

* The new spot cannot be placed on top of one of the endpoints of the new line. Thus the new spot splits the line into two shorter lines.

* No spot may have more than three lines attached to it. For the purposes of this rule, a line from the spot to itself counts as two attached lines and new spots are counted as having two lines already attached to them.

In the normal play, the player who makes the last move wins. Alternatively (misère), the player who makes the last move loses. Obviosly, the normal version is better for children.

Answer 4

Knot or not?

The topological game involves a projection of a knot, drawn onto paper, such as:

Player 1 picks an intersection and assigns a crossing (which segment of the curve is "above" and which "below" the other segment) and marks it on the drawing (for instance by making bold the "top" segment). Then Player 2 picks another intersection and likewise assigns a crossing. The players alternate until every crossing is assigned. Player 1's goal is to create the trivial knot (or "un-knot" or simple loop) while Player 2 tries to make any other knot.

When you're done, ask the child to test the final answer with a piece of string tied in a loop.

Even if the child doesn't quite know what strategy to employ for "winning," after the crossings have been assigned you can ask the child to "figure out" or "guess" whether the string would create a knot or not. This makes the activity more like an interactive arts and crafts exercise, with the ensuing delight of finding whether the string forms a knot. Pulling the string tight I imagine the child shouting in glee: "KNOT!" (or "NOT!")

(Start with a few very simple projections. Later, let the child draw the projection.)

Answer 5

There is an infinite indexed family of family-friendly, $\geq2$-player, perfect-information, draw-free-if-finite, cheap-to-construct, two-player combinatorial¹, solved, sequential games. They are easily formalizable, have considerable mathematical substance, easily suggest open problems (both in the direction of exact analyses feasible even for children, and in the direction of inventing mathematically-informed heuristics), and have connections to classical graph theory (from any given position, solving 'find a maximum matching' is sufficient to decide who has a winning strategy from this position). They seem to fit the

Question: Do you have some experience/proposals of "games" which you can play with your children, which would be on the one hand would make some fun for them, on the other would somehow develop their logical/thinking/mathematical skills, and on the other hand would be of at least some interest for adult mathematicians ?

rather well.

By the way,

I agree that 'Set' gives a child more opportunities for real-time, conclusive thinking, though quite repetitive conclusive thinking. Sadly, the path game, played on larger boards=graphs, feels like chess: though a perfect strategy must exist, one usually doesn't know it, and so one finds oneself reduced to heuristics. Likel in chess, even strong players must make arbitrary decisions, based on intuitive rules-of-thumb. What is mathematical about the PathGame is to think about it, less to play it. (Though one can beautifully demonstrate the relevance of the mathematical analysis of the game by playing on boards with an explicit maximum matching shown, such as the example boards below). The PathGame, too, needs careful explanation so as to not be misleading. (E.g.: a child might fancy themselves a 'master' of the PathGame, on the empirical evidence that they win each game, wielding some heuristics, but without having understood the principle.) Of course, repetition is important for learning. In a sense, it is deserved that 'Set' has most votes, in that it can be seen to fit the requirement of "develop[ing] their logical/thinking" skills very well.)

I now briefly describe the most usual² version.

Let $\mathsf{Graphs}$ denote the (proper) class of all symmetric irreflexive binary relations on a specified set. (Not necessarily finite, not necessarily connected.)

For each $G\in \mathsf{Graphs}$ let PathGame($G$) be the game defined by the following rules.

Rules of PathGame($G$). There are two players, 'b' (for 'blue') and 'g' (for 'green'). The players take turns making legal moves, and 'g' moves first. The player who first cannot legally move has 'lost'. Each move consists in nothing more than choosing a vertex of $G$ which (0) has not yet been selected by anyone, (1) is adjacent to the vertex selected in the immediately preceding move.

There aren't any other rules in the most basic version.

Needless to say:

In the beginning, condition (1), and of course condition (0), too, are void, so 'g' has a free choice of which vertex to pick first.

The simplest (and widely known) underlying mathematics I have summarized in the first non-quotation paragraph of my answer to this MO question.

I recommend that you try playing, understanding and varying the class function

PathGame:$\mathsf{Graphs}\to\mathsf{CombinatorialGames}$

with your child. There are open research questions related to this.

• PathGame is simple and mathematical. And it has a distinct geometrical/topological 'dimension' to it

• PathGame can easily and cheaply be 'realized' in many a 'medium'---even on a sandy beach when it's windy and the sand is not the finest.

• 'Building' instances of PathGame is easy (and easier than e.g. the implementation of some of the knot-games proposed, where in the end you'll need a high-quality, flexible rope or chain to do the 'testing' of the diagram). Playing it with pens only is probably not recommendable. Pens should be confined to the draw-the-board-phase. If you play with pens only, then it'll be one-game-per-board, while with movable tokens you can re-use a board. You should make the (small) effort of creating numbered 'tokens' (paper will initially do: you can simply use, say, ten specks of paper numbered in blue by 0,1,...,9 for one player, and ten further specks of paper numbered in green, again by 0,1,...,9. One player gets blue, the other green. This allows for at most twenty moves. Hence, with these tokens, you can 'play' all graphs with at most twenty vertices. (But most games will end long before all vertices are covered; note also that on all but the most trivial 'boards' it is possible to play badly, the game is not trivial.)

• On the other hand, it can be not-so-easy to lose the game intentionally even if you opponent tries to win (this can be seen as rather unusual; e.g., in chess, if your opponent tries to win, it is trivial to lose intentionally; though of course chess has also has the feature that if both players try to lose, then again it's not so easy to lose)

• This is a solved game, in a technical sense. There is a known (and even efficient, though this is not necessary to qualify as 'solved') algorithm to decide, from any legal position, which of the players 'g' and 'b' has a winning strategy.

• However, there is a non-trivial amount of 'preprocessing' you'll have to do to calculate a winning strategy Roughly speaking, if one allows, say, you to choose whether you want to play 'b' or 'g', then it'll take you (or your computer) time about $\lvert V(G)\rvert^3$ to do the preprocessing.

• Worth pointing out: once a maximum matching is known, by a preliminary calculation or by the grace of an oracle, then if it is a 1-factor (resp. not a 1-factor), then 'b' (resp. 'g') can effortlessly win even against an infinitely-intelligent and omniscient opponent. This does not go without saying, the point being that a winning strategy for e.g. a won-for-'g' Path game position has size (size of maximum non-perfect matching) + (small set of instructions) $\in$ $O(\lvert G\rvert)$, hence is rather small, while, for example, chess is different in this respect: it seems very unlikely that a winning strategy for (say) White in chess (it is not known whether there is one, of course) can be stated briefly. Now that is a fundamental difference between the PathGame and Chess: the PathGame admits of a relatively small winning-algorithm. (You can even take this as an occasion to discuss the concept of 'stored program computer's and 'a program is data' with your child.)

• In PathGame, if a relevant matching is known, then the player-which-has-the-winning-strategy can compute a (not: the) perfect response to any move of the opponent in constant time, more precisely, in one step.

• If your computer/mind/oracle only truthfully tells you with which 'player' (i.e. either 'g' or 'b') it'll be a win for you, but does not give you a relevant matching, then actually winning, even against your daughter playing randomly, will not be easy.

• $G$ need not be connected, though evidently the game will take place in one connected component of $G$ only; the initial choice of connected component which 'b' will have to make is another 'dimension' of this game.

• $G$ need not be finite; if it is, termination of the game is guaranteed by definition; if it is not finite, then the game may happen to run forever, though even then it is possible that the game ends (it is one of the variations and educational opportunities to try to analyze when this happens); it's even possible to manage to lose on a one-way infinite path.

• The numbers on the tokens I recommended above are not always necessary. More precisely, if you have zero short-term memory, then you'll need the numbers to be able to be able to decide whether you can still legally move. If you can remember the last move, no numbers are necessary to play legally. However, whether you will then be able to recover the path in the end, depends how far your memory reaches back into the past. With the numbered tokes, the path is 'stored' on the board an no remembering is necessary.

• Note that if one would weaken the rule for legal moves to 'any vertex not already selected, and adjacent to any of the already chosen vertices'.

Let us call this game PathGame${}_{t\mapsto t}$.

PathGame${}_{t\mapsto t}(G)$ is rather trivial: 'g' has a winning strategy if and only if $G$ has at least one connected component with a finite and odd number of vertices. For this trivial variation, the 'preprocessing' simply consists of counting the number of vertices in each connected component of $G$ (of course, if there are infinite components, even this trivial preprocessing may never end). And this variant game is also trivial in the sense that it is impossible not to win when first moving on a connected component with odd number of vertices.

• I noticed that the trivial variation just mentioned is just one end (the PathGame being the other end) of an infinite $\omega^\omega$-indexed 'spectrum' of similar games. For any function $h\in\omega^\omega$ and any $G\in\mathsf{Graphs}$ let PathGame$_h(G)$ denote the game with the same rules as PathGame($G$) except that a legal move at time $t\in\omega$ means to choose any not-already-chosen vertex which is adjacent to any of the last $h(t)$ selected vertices. In particular, PathGame($G$) = PathGame$_{t\mapsto 1}(G)$.

This is something of a weakening of the 'directionality' with which the PathGame unfolds.

I do not know whether e.g.

• PathGame$_{ t\mapsto \log t}$

• or even PathGame$_{ t\mapsto 2}$, ,which I recommend your child may study,

have been analyzed in the research literature.

Note also that this way we have defined $\omega^\omega$-many, at least intensionally-distinct (though extensionally often rather similarly-behaved; evidently all $h$ which grow faster than the identity behave the same; also, non-monotonic $h$ will probably considered to be bizarre by many) boolean valued graph invariant. For each $h\in\omega^\omega$ let $\eta_h\colon\mathsf{Graphs}\to$ $\{$ $\perp$, $\mathsf{T}$ $\}$ denote the predicate which, given any graphs $G$, returns $\perp$ if 'b' has a winning strategy in PathGame$_h(G)$, and returns $\mathsf{T}$ if 'g' has one, and returns $\mathsf{T}\hspace{-1em}\perp$ if neither 'g' nor 'b' can force a win for themselves. (The latter evidently can happen only if $G$ is infinite.)

Then $\eta_h\colon\mathsf{Graphs}\to$ $\{$ $\perp$, $\mathsf{T}$, $\mathsf{T}\hspace{-1em}\perp$ $\}$ is manifestly an isomorphism invariant function on $\mathsf{Graphs}$.

We know that

• $\eta_{t\mapsto t}$ is a predicate which corresponds to whether a given graph has at least one connected component with a finite odd number of vertices

• $\eta_{t\mapsto 1}$ is a predicate which tells us whether a given graph has a perfect matching

Moreover, there are $\aleph_0^{\aleph_0} = 2^{\aleph_0}$ such graph-invariants, and they are all 'intensionally distinct' (though probable many of them are 'extensionally' rather indistinguishable).

There are also finer, i.e., non-(booelan-valued), graph invariants that PathGame gives rise to.

• I don't know whether this has been analyzed in the literature so far.

• So, in particular:

These are games which your child may 'grow into', with time learning different concepts like 'odd', 'infinite', 'perfect matching', 'maximum matching', 'maximum/maximal', 'graph invariant', 'truth values', 'intuitionism', 'complexity of computing a maximum matching',... She might even once publish the best study of the class-function PathGames:$\mathsf{Graphs}\to\mathsf{CombinatorialGames}$ so far. (A realistic first step: 'solve' PathGame${}_{t\mapsto 2}$.)

There are many independent 'dimensions' of interactivity and freedom:

• choice of the board=graph $G$ (though this choice is somehow illusory: one can define it away by letting there be only one large board, consisting of the large-in-the-technical-sense graph $\coprod_{G\in\mathsf{Graphs}}G$. Then the 'building the board'='choosing the first vertex in the big board'.

• who gets to choose who plays first,

• what heuristics are there if one decides to want to let the other win, i.e., decides that one wants to lose?

• whether to tell her the 'secret' behind how to 'win' this game or do you let her find out herself,

• PathGame 'compositions' (as in: 'chess composition'), with partially filled boards, and an instruction saying something like " 'g' to move and win "

• three players, while otherwise retaining the rules of the 'classical' version PathGame${}_{t\mapsto 1}$. The three-player version seems not to have been analyzed so far. I did not think about it. With three players, new difficulties³⁴ arise, in particular with regard to possible collusion among two of the three players (and there are more sub-dimensions here: do you allow the players to freely communicate, and e.g. agree on tactics, or is all the information available to them the information they see on the board?

• ....

• Another dimension is memorizing a board, and then playing the game without a board, for example playing the PathGame while taking a walk, or over a phone. (There are many aspects to this; and this a field of research of its own, e.g. search for information about playing poker over the phone.) I think one cannot conveniently play 'Set' over a phone. I recommend the 22-vertex graph given below, which is a win for 'g', for memorization: it is neither trivially small, nor difficult to memorize, especially with the symmetries and the labelling-rationale that I present below. A worked example for such a phone-play on the 22-vertex graph given below, relative to the labelling given below, would be the 'dialogue', in which 'g' plays according to the maximum matching shown in red in the version which uses Babylonian degrees, and in which the choice of the 'non-matched' vertex ${\huge\text{$\frac{\color{green}3}{\color{green}4}$}\pi}\ \text{$\frac{\color{green}4}{\color{green}5}$}\pi$ in the beginning is an arbitrary choice among the vertices left non-matched by the red matching.

  $\Huge``\quad$ ${\Huge\text{$\frac{\color{green}3}{\color{green}4}$}\pi}\ \text{$\frac{\color{green}4}{\color{green}5}$}\pi$ $\Huge,\quad$ $\large\mathrm{{\color{blue}s}}$-$\large\mathrm{{\color{blue}p}}$ $\Huge,\quad$ ${\Huge\text{$\frac{\color{green}1}{\color{green}4}$}\pi}\ \text{$\frac{\color{green}6}{\color{green}5}$}\pi$ $\Huge,\quad$ ${\Huge\text{$\frac{\color{blue}1}{\color{blue}4}$}\pi}\ \text{$\frac{\color{blue}8}{\color{blue}5}$}\pi$ $\Huge,\quad$ ${\Huge\text{$\frac{\color{green}1}{\color{green}4}$}\pi}\ \text{$\frac{\color{green}4}{\color{green}5}$}\pi$ $\Huge,\quad$ $\large\mathrm{{\color{blue}n}}$-$\large\mathrm{{\color{blue}p}}$ $\Huge,\quad$ ${\Huge\text{$\frac{\color{green}5}{\color{green}4}$}\pi}\ \text{$\frac{\color{green}6}{\color{green}5}$}\pi$ $\Huge,\quad$ ${\Huge\text{$\frac{\color{blue}5}{\color{blue}4}$}\pi}\ \text{$\frac{\color{blue}2}{\color{blue}5}$}\pi$ $\Huge,\quad$ ${\Huge\text{$\frac{\color{green}5}{\color{green}4}$}\pi}\ \text{$\frac{\color{green}4}{\color{green}5}$}\pi$ $\Huge,\quad$ ${\Huge\text{$\frac{\color{blue}5}{\color{blue}4}$}\pi}\ \text{$\frac{\color{blue}8}{\color{blue}5}$}\pi$ $\Huge,\quad$ ${\Huge\text{$\frac{\color{green}5}{\color{green}4}$}\pi}\ \text{$\frac{\color{green}0}{\color{green}5}$}\pi$ $\Huge"\quad$, at the end of which 'b' knows he has lost.

• You can experiment with rules like 'your child is allowed to design the playing field, yet you are then allowed to choose who plays first. Or vice versa (you design, your child chooses who plays first). Or even: your child decides who decides what, and from then on, everything must follow the logical rules. And then there is a complexity dimension. Even assuming you understand the whole game better than your child, and (say) you play the child-creates-the-board(=graph),you-decide-who-moves-first version, you will have to compute whether the graph your child drew for you has a 1-factor or not, and this is, while well understood, not easy to do, especially mentally. And you can learn much about algorithms for finding 1-factors while still playing with your child. This game is simple, variable and inexhaustible.

• Ending on a utopian note, one can imagine that you play variations of this game with your daughter all your life, possibly over the phone if the two of you agree upon a 'memorized' board (e.g. the 22-vertex board I gave above; remember, it's a 'win' for 'g'). And she might try 'solving' some $>2$-player variants. If she is older, or maybe even now, another dimension could be that she teaches a machine to play this game, or she programs a machine to do the 'compute a maximum matching' calculation, or even (assuming technology advances further) that she programs a robot with a camera to do these calculations after 'sight-reading' (so to speak: the point being that the robot is not allowed to have the board-plus-strategy stored into it, just like a sight-reading musician does not have the score stored in their memory) the board(=graph).

Ready-to-play boards for the PathGame.

Here I give some explicit 'boards'='graphs', roughly in order of increasing difficulty of playing on them. Some come complete with maximum matchings included. Some don't have a matching shown.

The small boards are not labelled. For the large boards, I use a consistent principle for labeling the vertices. The principle is self-explanatory and can also serve to discuss angles. Decoding the rationale for the labelling could be another (trivial) educational aspect of all of this. There is one small variation: sometimes I used 'Babylonian' notation, sometimes I use 'fractions of $\pi$'.

The 22-vertex graph given last is not planar, since it contains (many $K^5$-minors, yet it can 'almost' and rather apparently-rather-naturally be immersed (not: embedded) into the 2-sphere, with four crossings only. Just take the 'north-pole' and 'south-pole' notation below as a hint for how to visualize that. It might make for a nice game to have this graph actually realized on a washable spherical surface. (Incidentally, I don't know whether $4$ is its crossing number, yet I conjecture it is.)

There is another dimension:

Realise the 22-vertex graph with four-crossings on the surface of a 2-sphere. Perhaps it will be a better use of the internet to ask someone who is adept in 3D-printing to do this.

(Incidentally, I don't know whether the crossing number of this 22-vertex graph is 4. This is relevant to the OP since a good 'realization' of this game, not overly confusing for children, should be drawn as simply as possible.)

On washable displays, it is possible to play PathGame${}_{t\mapsto 1}$(the graph represented by the respective picture) with two non-permanent markers, one green, the other blue. Please don't try this if you are in doubt about at least one of the following: (0) the washability of your display (many aren't really washable, or at any rate not made for being wiped often) (1) the 'aggressiveness' of the marker-ink you are using. Printing, or playing by means of some graphics software, seems safe.

'b' to move second and win:

'b' to move second and effortlessly-win:

'g' to move first and win:

Notes on this graph. This is, in a sense, a⁷ smallest cubic graph on which 'g' can force a win; see my comment to this MO question. Note however, that if 'g' wants to lose, 'g' can force a loss of 'g', and 'b' cannot make 'g' win then. ⁵

'b' to move second and win:

'b' to move second and effortlessly-win:

The above two boards illustrate one of the may aspects of the PathGame: this is graph which is easily shown to have even a perfect matching, but when playing on a plain, unmarked board, it is not easy for 'b' to actually choose one fixed matching, and keep it in mind to let their moves be guided by it.

'g' to move first and win:

Notes on this graph. In a sense, this is the smallest board=graph in which every vertex has four neighbors and in which 'g' has a winning strategy. Recall that for 'g' to have a winning strategy, it is necessary and sufficient that there does not exist a 1-factor. By the ($r=4$)-instance of Corollary 2a in Gary Chartrand, Donald L. Goldsmith, Seymour Schuster: A sufficient condition for graphs with 1-factors. Colloquium Mathematicum. Volume XLI, Fascicle 2, 1979., every 4-regular, edge-2-connected graph $G=(V,E)$ with $\lvert V\rvert$ even and $\lvert V\rvert < 4^2+2\cdot 4-2 = 22$ has a 1-factor. The contrapositive of this implies that if you need a 4-regular edge-2-connected graph without a 1-factor, then you'll have to use 22 vertices or more. Please note: op. cit. does not seem to prove that the above graph is up to isomorphism the only *4-regular edge-2-connected 22-vertex graph without a 1-factor. It might be another possible project related to PathGame to extend the results of op. cit. in the direction of proving the (non-)uniqueness of the relevant extrema.

'g' to move first and effortlessly-win:

Notes on this illustration. The red edges indicate a maximum matching $M$. Proof: the set $\{\text{n-p},\text{s-p}\}$ is a 'bad set' in the sense of Tutte's good characterization of the class of graphs with perfect matchings, since it has 2 elements, yet deleting it and all incident edges from the graph leaves 4 odd components. Thus, $G$ does not have a 1-factor. Hence each matching in $G$ has at most $\frac12\lvert G\rvert - 1 = 10$ edges. The ten red edges achieve that upper bound. This proves that $M$ is maximum.

Furthermore, the 'effortless'(=computable in one step from the given board) winning strategy for 'g' is to choose one of the two unmatched vertices, and henceforth *always let the 'response-move' to 'b' 's move be the unique other end of the relevant matching edge; if there would ever come a step at which no such response move would be available, then this would imply the existence of an augmenting path, which is however impossible because of the matching being maximum. Therefore, 'g' will always have another move-along-an-M-edge. Since the graph is finite, there must come a time when b does not have another legal move.

Footnotes

¹ An introduction to the research literature is e.g. Aviezri S. Fraenkel: Combinatorial Games: Selected Bibliography with a Succinct Gourmet Introduction. The Electronic Journal of Combinatorics (2009), #DS2

² I think the easiest version is PathGame${}_{t\mapsto t}$. The 'preprocessing' then amounts to 'mere' counting, which may be challenging enough at age five. Note that for any $h\in\omega^\omega$ with $\forall t\quad t\leq h(t)$, PathGame${}_{t\mapsto t}$ is 'extensionally the same' as PathGame${}_{h}$.

³ This seems the most intuitive convention; it is evidently similar to 'stalemate' or 'being checkmated'; it is very dissimilar to 'having less cards'.

⁴ James Propp: Three-player impartial games. Theoretical Computer Science. Volume 233, Issues 1–2, 2000, Pages 263-278

⁵ Someone who has seriously worked on three-player combinatorial games seems to be Katie Doles.

⁶ How can 'g', as always moving first, nevertheless force its own loss, even against a 'helpful' (or more clearly put: against any strategy of 'b') player 'b'?

⁷ Again, it seems not to be known whether this is the only isomorphism type of 22-vertex 4-regular edge-2-connected graphs without a 1-factor.

Answer 6

Dots and Boxes

Is a pencil-and-paper game for two players. It's quite simple to explain but quite hard to play. Five year olds should be able to learn it and with some training maybe also being good at it.

Go

At least same simple stuff like trying to catch something (first one to catch anything wins) on a small board. If you feel you are not challenged enough, just give your child some extra tokens at the beginning.

Spookies

Board game featuring addition and subtraction until 12 (two dices) as well as a bit of expectation calculation. They say it's from 8 but we started playing from 6 and it went well.

Phutball

A two-player strategy board game can be played on a Go board and with Go tokens. It's sufficiently hard for computers, so it may be hard for you too.

Draughts, Reversi

Classic strategy games with simple rules but not too simple strategies.

Matchstick puzzles

Do them first and think about new ones.

Logic puzzles

You would need to find a collection, book of interesting logic puzzles but they are usually quite a lot of fun. A classic is the wolf, goat, cabbage all need to go over a river but only one item can go into the boat and the wolf eats the goat, the goat eats the cabbage if left alone.

Solving problems of the Kangaroo test

Popular fun math test for pupils from first class in school (from 6 years on). Test for the last 17 years are available online in English for download. Print them out (the ones for the lowest age level), mark the ones you think are adequate, then explain them to your child.

Example: Old McDonald has a horse, two cows and three pigs. How many more cows does he need, so that exactly half of all his animals are cows?

Answer 7

What about Spot It! (US), also known as Dobble (Europe)?

We are given a deck of 55 cards. Each card contains 8 different symbols, such that any pair of cards in the deck has precisely one symbol in common. There are various versions of the game, all based on speed and pattern matching. The advised age group is 7+, but boardgamegeek.com rates it as suitable for 4+.

The game, and the finite projective geometry behind it, has been discussed on Math.SE and especially Stackoverflow. See also Dobble-et-la-geometrie-finie (in French), from which the following picture (for an example of a deck of 7 cards) comes:

          

Answer 8

The following game is fun if you are:

• a) In the UK,
• b) In the car,
• c) Trying to improve your child's mental arithmetic.

Pub Legs

Assuming that you have two children, one on each side of the car. Whenever you pass a Public House the child on that side of the car has to calculate the sum of the number of legs of any creature mentioned in the name and add it to their total. For example, The Dog & Duck would be worth 6 points. The Ploughman would be worth 2 and so on. Adjudication from the front seat would be required for The Fox & Hounds (i.e. how many hounds are on the pub sign). The child with the highest score at the end of the journey wins. To avoid arguments, maintain the total for the return journey, which assuming that you go back the same way that you came and everyone is paying attention, guarantees a draw.

Answer 9

I invented a math game a couple years ago called The Chaos of Operations (a play on the order of operations), and while the rules as written are intended for people with college mathematics and/or programming knowledge, it is easy to strip a few rules and make it playable for young children. You can even reintroduce a few rules at a time to take it from a first-grade level game all the way through the aforementioned college level.

I have a web-playable version of the game here, and pictures of the physical board game here.

I'll first describe the rules of the more advanced version, and then follow with what to do to simplify it for children. The instructions had lots of pictures to help explain things, and I don't have access to those assets as I'm typing this answer. I suggest playing the web version of the game to see for yourself how it works if my words are unclear.

Setup:

Each player gets 10 blank cards (in the physical prototype, they are cardboard with masking tape so I can write/erase with a dry-erase marker). You will populate the cards with numbers, and which numbers you use will depend on your desired difficulty.
Easy: Each player gets the numbers 1-10
Medium: Roll a 20 sided die 10 times. Each player gets a copy of each number rolled. (The web version of the game linked above uses this rule).
Hard: Roll two 10 sided dice, one marked with 1-10 and one with 10-100 (in intervals of 10), and sum them. This gives you a value from 1-100. Repeat this process 10 times total, giving each player a copy of each number created.

Each player also gets 10 operator cards. Both start with 2 + cards, 2 - cards, 2 ×, 2 ÷, 1 ^ (exponent), and 1 mod (modulus).

Both the operator and number cards are placed on the table in front of each player. Both should have an identical set of the 10 operators, and 10 numbers chosen via your difficulty settings.

Finally, place two cards with the number 1 in the play area, which is on the table above each player's hand. Each card is placed in front of each player.

There is a last-turn advantage here, so the player with the most mathematical knowledge moves first.

Objective

You are going to be using your numbers and operators to modify equations, taking the order of operations into account. At the end of 10 turns when each player has exhausted their number and operator cards, evaluate the equations. The player with the largest number wins.

Play

On your turn, select one operator card and one number card from your pile. You may place the two of them, sequentially, in either player's equation. The only rule for placement is that the equation must still be valid. Indefinite form is allowed to exist during play, and is handled specially if it still exists at the end of the game.

Example:

First turn, both player's have the same equation: 1. The first player decides to play a + and a 10 to make their own score larger. At the end of the turn, player 1's equation reads 10+1, and player 2's still reads just 1. Player 2 can choose to make their own score bigger, or player 1's score smaller. Say they choose the latter. They can take a 1 and a - card from their hand, and add them to player 1's equation, so it reads 1-10+1, while their own equation still reads 1. At this point, player 1's score is -8, and player 2's score is still 1. Repeat in this fashion for 9 more turns.

Example:

Player 1's equation reads 1+5-6x2. Making their score -6. They would like to make their own score larger, so they decide to insert a 1 and a +. The optimum move is to insert them so their equation reads 1+5-1+6x2, changing their score to 17. Remember that when inserting, the equation must still be valid. They can't arbitrarily insert the operators and elements, so 1++15-6x2 is an illegal move. Numbers cannot be concatenated by being placed sequentially, because that would leave illegal operators (yes I know, ++ is legal in programming. This isn't programming :P).

Winning

When all 10 turns are exhausted, evaluate both equations. Don't forget to use the order of operations during this evaluation. If one of the equations contains indefinite form something/0 or 0^0, their equation is to be treated as -infinity. The highest number wins.

Note:

The board game version has extra rules for adding parenthesis, swapping numbers, and swapping operators. I've ommitted them here for simplicity. If you want to play this game yourself and would like to know what they are, please comment below.

For younger audiences

You can omit or modify some of the starting resources to simplify this game for children, depending on their level.

• the random numbers 1-100 are probably too much, so use the numbers 1-10. If you want a little randomness still, you can choose them with a 4, 6, 8, or 10-sided die.
• exponent and modulus operators are hard to play around for people not familiar with them. You can omit them, and replace them with additional + or - operators.
• you may want to take fewer turns than 10, to ease with the mental calculations during play.
• if the child is in elementary school but hasn't learned multiplication and division yet, they may be replaced with additional + or - operators.
• if you want to exclude the possibility of negative numbers entirely, you can use just the + operators. However, most of the strategy of the game is lost if you do this.

Some things I learned from watching people play

It seems like this game would be very dry, and lots of number crunching, however it has far more appeal than I anticipated. I have a lot of programmer friends and they especially love this game, but when my less mathematically-inclined friends and family play, it still turns into very intense and competitive games. The strategy lies in the fact that you don't usually need to know the exact value of your or your opponent's equation, as you can play just off of the patterns in the operations. For example if your opponent has something giant like 46^78 in their equation, you don't need to know what that is. You just need to know you can get rid of it by dividing right before, so you might play 12/46^78, making their giant asset now a tiny positive number very near 0. That player might respond by using a + or - to break up the evaluation order, so they might retaliate with 12/16+46^78. You might retaliate again by using modulus, 12/16+46^78 mod 17. At no point during this example did I bother figuring out what 46^78 is, I just know that I can use division and modulus to take a big threat and then them into a small one, and plus or minus to break up undesirable operations. The strategy in the game comes from knowing these tricks so you don't actually need to do the math until the very end.

Answer 10

I've written a game that's a mix of algebra and maze solving. I believe its pretty relevant to this question. There are easy levels but also it can get surprisingly complex even for small mazes. Its called Numplussed and its free on Android or iOS:

Numplussed for iOS

Numplussed for Android

Some medium/easy levels:

Example of a hard level:

https://twitter.com/codeulikegames/status/911725658929291264

Answer 11

Bennett's pebble game is a one player game. The rules for Bennett's pebble game are as simple as possible. Furthermore, all that is needed in order to play this game is a pencil, a sheet of paper, and some pebbles/coins.

The board for this game shall have $n+1$-spaces which you can draw by hand. The spaces are labelled from Space $0$ to Space $n$. In the game, you initially have a supply of $m$ pebbles where $m\ll n$.

In the initial state of the game, space $0$ has one pebble on it and no other space has a pebble on it. The goal of the game is to end up with a pebble on Space n and a pebble on Space $0$ and where none of the other spaces have pebbles on them (i.e. the goal is to have only two pebbles on the board with one at the beginning of the board and the other at the end of the board), and the objective of the game is to achieve this goal in as few moves as possible. There is only one rule to this game. If $i>0$, then you may add a pebble or remove a pebble from Space $i$ if and only if there is a pebble on Space $i-1$. Now, this game is challenging because you only have a limited number of pebbles.

This game arose in Charles Bennett's investigations of time/space trade-offs for reversible computation where he has calibrated how well a reversible computer can emulate conventional irreversible computation. In this game, the number $m$ of pebbles represents the space available for computation, the number of moves taken represents the time needed to perform the computation, the number $n$ represents the number of steps needed to perform the computation using a conventional irreversible computer, adding a pebble represents computing the next step in the computation, and removing a pebble represents uncomputing the next step in the computation. An optimal (and not overly complicated) strategy for Bennett's pebble game has been found in this paper. There are generalizations [3][4] of Bennett's pebble game including a version where one is allowed to remove a fixed number of pebbles without having a pebble on the previous space (this generalization represents nearly reversible computation where a small amount of data is allowed to be deleted).

[3] Ming Li and Paul Vitanyi. Reversible simulation of irreversible computation. In IEEE Conference on Computational Complexity (CCC), 1996.

[4] Ming Li, John Tromp, and Paul Vitanyi. Reversible simulation of irreversible computation. Physica D, 120(1):168–176, 1998.

Answer 12

Ultimate tic-tac-toe is played on a two-level hierarchy of tic-tac-toe boards. The rules are simple enough for a young child, but the strategy is pretty complicated.

Answer 13

The racetrack game might not be as much of a true math game, but it does require arithmetic to play, demonstrates the concepts of inertia, acceleration and deceleration, and depending on how you present it can include crashes and explosions, with nothing but graph paper and pencil.

Making relevant sound effects and age-appropriate trash talk part of the game and it can be a lot of fun. If cars aren't of interest, you can make the race be between frogs, dragons, penguins, or shopping cards. Enhance the game by adding secondary objectives (stopping to get fuel/buy ice cream/whatever for example.)

Answer 14

I like the 'four fours' game, it's simple but challenging for a child. The idea is to use four 4s in any way in algrebraic operations to calculate all numbers from 0 to 9. All four 4s must be used.

For example:

0 = (4-4)+(4-4)

1 = (4/4)+(4-4)

etc... up to 9

Answer 15

I often play the game Doubled, Squared, Cubed! with my kids, as I did as I child myself years ago with my siblings. It can be played with kids of any age, and it is a great way to expose the kids to new mathematical operations that they might not yet encounter in school.

Read about it on my blog (link above), but here is a sample play to give you the idea. The agreed range here was integers in the interval $[0,100]$. Now that the kids are older, we usually have a bigger range, including negative numbers.

Hypatia: one

Barbara: doubled

Horatio: squared

Joel: cubed

Hypatia: plus 36

Barbara: square root

Horatio: divided by 5

Joel: times 50

Hypatia: minus 100

Barbara: times 6 billion

Horatio: plus 99

Joel: divided by 11

Hypatia: plus 1

Barbara: to the power of two

Horatio: minus 99

Joel: times itself 6 billion times

Hypatia: minus one

Barbara: divided by ten thousand

Horatio: plus 50

Joel: plus half of itself

Hypatia: plus 25

Barbara: minus 99

Horatio: cube root

Joel: next prime number above

Hypatia: ten’s complement

Barbara: second square number above

Horatio: reverse the digits

Joel: plus 3 more than six squared

Hypatia: minus 100

and so on!

One must never say the number exactly, but prove that you know it by providing an operation that makes sense with that number, but not with many other numbers, or that makes the new value arrive at a famous number. One might, for example, often bring the number back to a specific value such as $17$ as a way of proving that you still know what the number is.

Answer 16

There's a commercially available game called rat-a-tat cat which teaches things about probability and reasoning rather nicely. It can be played with ordinary playing cards as well, although the game-cards have silly cat pictures that make them fun. The idea is to assemble a group of 4 high-value cards. You get dealt 4 cards (face down), which I'll call your "line" and get to "peek" at the outer 2 at the start of the game.

Players take turns drawing a card from the deck, looking at it, and then either keeping it, and discarding one of their own cards, or simply discarding it directly. The discard pile is face up. A player may also, if the discard pile is nonempty, pick up the top card from there (if it's not a face card) to swap with one of their own cards.

In the table center are two piles: the remainder of the deck, face down, and the discard pile, face up.

A turn generally consists of drawing a card from the top of one of these two piles (the player decides which), and then either discarding it or replacing a card in the player's "line" with the card, and discarding the line card.

Play rotates clockwise. At any time when a player thinks that their hand might be better than those of other folks, that player, when it's their turn, can say "rat a tat cat!" and complete their play. Each other player then gets to play once more, and then everyone shows their cards. The one with the highest total wins. The total is based on JUST the number cards A, 2, 3, 4, 5, 6, 7, 8, 9.

The fun part of the game comes from Jacks, Queens, and Kings, which only have their special power when drawn from the DECK, not from the discard pile. I'm going to get the roles wrong here (our kids are now in college1) but let me do my best. A Jack lets you peek at any one of the 4 cards in your line. A Queen lets you exchange any one of your line cards for any one of another player's line cards, without looking at either. A King lets you play up to 2 more times: first they pick the next card from the deck. If they decide to swap it for one of their line cards and discard that line-card, the turn is over. If they decide to discard it, however, they get to draw again from the deck and either use that card to replace a line card, or decide to discard it as well.

When the "deck" runs out, the discard pile is shuffled and turned over to become the new "deck".

Strategy:

If you see your friend "keep" a drawn card, you might figure it's a good one. When you draw a Queen, you exchange the "2" that is your left-hand card for that player's supposedly good card.

What happens in the course of play is that players gradually learn what all four of their cards are (Jacks help!), and perhaps learn something about the opponents' hands as well, thus allowing them to guess the ideal moment to cry "rat-a-tat cat".

When playing adult-vs-child, you can let the kid look at all four face-down cards at the start of the game, while the adult gets to look at only one or two; this knowledge is a substantial advantage. (If the kid sees 9-9-10-10, they can cry rat-a-tat Cat on their very first move, because the adult is very unlikely to have such a good hand!) You can also have a rule where for the kid, a Jack means "peek at one line card in ANYone's line" rather than just the kid's own line.

It's a fun 3- or 4-person game, but works surprisingly well with only two players. I can't recall the rule for what happens with a King when you draw a face card rather than a number card. If it's a Jack, and you use it to look at one of your middle cards, and then discard it, does that count as having used it, so the turn ends, or not? Again, I suppose that as a means of handicapping, you might make one rule for kids and one for adults.

Scoring: the person with the highest score wins (or the lowest score -- I suppose you can pick your variant). For multi-game sessions, keeping a running tally is more fun that just a list of how many times each person won.

Answer 17

DragonBox is a great mobile app for all ages. It teaches the principles of algebra quite effectively, and someone who isn't familiar with algebra doesn't even realize that they are learning math until several hours into gameplay.

Answer 18

• Dots and boxes
  Easy rules. Tough to always win. If you want to make the game really tough, make the board small!! eg: 4x5.
• Tic Tac Toe
  The waiting moves are worth attention for those who are not new to the game.

• River crossing puzzles
  For kids

  • Fox, goose and bag of beans
  • 3 missionaries and 3 cannibals

  For adults

  • Bridge and torch problem
  • Jealous husbands problem
• Chomp
• Nim
• Sprouts
• Show cutting a normal strip and Möbius strip

--------------edit added---------
Many games in the first two books of "Winning Ways for Your Mathematical Plays" would score a point here. I have mentioned chomp,nim and sprouts above. Simple subtraction games would be good for little kids. A sample game is the following. We have a number of pebbles. A player can take one or two pebbles. The players take turns and the player who takes the last pebble wins.

By the way whichever game we choose, it would be a good idea to introduce the game in its simplest form (with minimum number of rules) first and then may be gradually increase if they are so much interested in the game.

Answer 19

To answer the part of your question about games that can be used to teach math concepts: Zombie Dice! (or any of their spin-offs)

I've always thought this would be a GREAT way to teach thinking about probability in a useful way.

The goal is to eat brains and not get shot. Its is a dice rolling game where there are three colours of dice: green (most brains, least guns), yellow (less brains, more guns), red (mostly guns, few brains). The dice are pulled "blind" from a cup and then rolled. The side facing up shows what you got: Brains, a re-roll, or if you got "shot". You keep pulling dice and rolling until you choose to pass your turn or accumulate three shots. When you pass your turn, you keep the brains you rolled as points. When you accumulate three shots, your turn is over and you LOOSE all the brains you accumulated.

How probability comes into play: There are specific numbers of each dice, so you know if you successfully rolled two red dice and a yellow one, that there is a high probability of pulling green dice from the cup which are the least risky in terms of being "shot". If, on the other hand, you have rolled all the green and many yellow dice and have been "shot" twice - you have BAD chances remaining as you will likely pull a red dice and that is very risky to roll as it usually yields being "shot".

Answer 20

Pokematrix is a game I came up with to play my son's (6 years old) pokemon card collection. Despite what it says on the packaging, pokemon is not suitable for kids that young, I would guess 10ish to play properly.

You can work out the game from the image. Play random cards, high level beats low level, same level check the graph. It looks complicated but a 6 year old can follow it.

You can add a layer of strategy by building a hand from a pool of cards.

Answer 21

A board game called The Siblings Trouble ($25) was released in 2015. It is meant for families and includes elements of story telling and math. I looked through a box of it last night and was impressed at how accessible the game seems for a younger audience. By emphasizing story telling and math I imagine children will be engaging their critical thinking skills. I am a big fan of mentally engaging children as much as possible so the idea of this game is attractive.

For example one card called the Action Figure allows a player to add a point to a skill, but only if the player is able to describe how using the action figure would be able to help them in their story.

Answer 22

i create a simple play with the 4th basics operation here : Jeux de calcul

The game is on french, but you can use the translator. In this game you could choose a "bonus" and sublimes gifs and images from space append when you growth your score, you can imprim or recive by e-mail your results, e-mails are not recorded, no publicity, easy to play simply good for learning basics maths. I created this play for my child. Bye !

Answer 23

I've been writing little one-page javascript pages to introduce my four- and six-year-old to various mathematical concepts like cardinality, place-value, sets, factors, equivalency, etc, etc:

http://ideonexus.github.io/Explorable-Explanations/

The boys love some of them (others not-so-much). Most of the code is original, but I try to be careful to give credit to anyone whose code or ideas I build upon.

Answer 24

You can play Tic-Tac-Toe on the affine plane of order 5. Check up PentacTow.

---

I have a few further ideas for applications gaming finite geometries and I take this opportunity to invite whoever codes and have interest in such a project to contact me (the TicatacToe was programmed by my brother, Gal Bader, but he is too busy now).

Answer 25

A game I loved as a kid in elementary school was Number Munchers. There are a bunch of variations around, including an Android version, so Google it.

https://play.google.com/store/apps/details?id=com.UTW.GameNC&hl=en

https://i.stack.imgur.com/wYDIE.gif

Ok, so that's an Android version, plus a Gif of the original game. I played it on an old Apple II series computer.

The game gives you options for super basic arithmetic as well as slightly higher math, such as knowing prime numbers, as shown in the Gif.

There are bad guys and timed levels, so it has to be one of the first math gamification games around.

I liked it better than Oregon Trail, so I hope you enjoy it too!

Answer 26

I would like to just share an idea.

If you would like to develop her arithmetical skills, then how about Thousand the card game?

To be a good player, you should be able to estimate how many points you are able to gather, based on cards which you hold on hand.

Answer 27

Most card and board games at best only need simple addition. Here is a variant of the card game Blackjack (or 21) which requires both addition and multiplication. Only use cards with values up to N, where N may be 10 for an older child, and say 5 for a younger child. Let's say N=5. Then the goal is to achieve the highest number up to 25, without going over. Play: First deal two cards, say 3, 4: then your score is 7. The player can hold or ask for another card, say she receives 2. Then she has three possible scores: 2+3*4=9, 3+2*4 = 11, or 4+2*3=10. If she asks for another card, she then divides her cards into two pairs, multiplies the cards within each pair, and sums the total. Thus if the cards are now 2,3,4,3 a possible score is 2*3 + 4*3 = 18. Keep on going, dividing your hand into pairs and (if an odd number of cards) a singleton. You are out if all possible pairings lead to a score greater than 25.

Answer 28

Possibly a bit much for a five year old, but Penrose tiling? With coloured card and round-nosed scissors, of course...

Answer 29

There are many interesting math games that anyone can play in Berlekamp's "Winning Ways for your Mathematical Plays" and Martin Gardner's books, such as the "Colossal Book of Mathematics."

For example, how about Hackenbush? To play, first draw a figure (such as a person) on a whiteboard, using two colors of fully-connected line segments, say red and blue. The figure must be connected to a neutral "ground" line. A red player and blue player take turns erasing one segment of their own color, along with any segments that become disconnected from the rest as a result. The first player without a legal move loses.