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6
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edited Jul 25 2011 at 10:02 |
Story
I was bored sitting in front of my computer and using a rectangle to select icons on my screen. I could select $1$, $2$, $3$, $4$, but not $5$ icons.
(Black squares are the icons. Note that it is possible to find rectangles with $1$, $2$, $3$ ad $4$ black squares in respectively.)
So I rearranged the icons into the following arrangement:
So in this arrangement, there exists a rectangle that includes $i$ black squares, where $1 \le i \le 6$. However, on the other hand, $6$ is not the maximum. For example, we can actually achieve $7$ with this arrangement.
Problem Formulation
$a$ is a $n$-iconic number if there exists an arrangement function $f:\{ x \in \mathbb{Z} | 1 \le x \le n\}^2 \rightarrow \{ 0, 1\}$, such that for $1 \le i \le a$, there exists a quadruple $(\alpha_i, \beta_i, \gamma_i, \delta_i)$ such that
$$\sum_{j = \alpha_i}^{\beta_i} \sum_{k = \gamma_i}^{\delta_i} f(j, k) = i$$
Find the maximum $n$-iconic number.
If $a$ is an iconic number with an arrangement function $f$ with an additional criterion
$$\sum_{j = 1}^n \sum_{k = 1}^n f(j, k) = a$$
then $a$ is $n$-perfect iconic
Is the maximum $n$-perfect iconic number the same as the maximum $n$-iconic number?
Find the maximum $n$-perfect iconic number.
Small Cases
For $n = 4$, the greatest iconic number and the greatest perfect iconic number is $12$, by the following construction:
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5
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edited Jul 25 2011 at 2:09 |
Story
I was bored sitting in front of my computer and using a rectangle to select icons on my screen. I could select $1$, $2$, $3$, $4$, but not $5$ icons.
(Black squares are the icons. Note that it is possible to find rectangles with $1$, $2$, $3$ ad $4$ black squares in respectively.)
So I rearranged the icons into the following arrangement:
So in this arrangement, there exists a rectangle that includes $i$ black squares, where $1 \le i \le 6$. However, on the other hand, $6$ is not the maximum. For example, we can actually achieve $7$ with this arrangement.
Problem Formulation
$a_n$ a$ is a $n$-iconic number if there exists an arrangement function $f:\{ x \in \mathbb{Z} | 1 \le x \le n\}^2 \rightarrow \{ 0, 1\}$, such that for $1 \le i \le a_n$a$, there exists a quadruple $(\alpha_i, \beta_i, \gamma_i, \delta_i)$ such that
$$\sum_{j = \alpha_i}^{\beta_i} \sum_{k = \gamma_i}^{\delta_i} f(j, k) = i$$
Find the maximum $n$-iconic number.
If $a_n$ a$ is an iconic number with an arrangement function $f$ with an additional criterion
$$\sum_{j = 1}^n \sum_{k = 1}^n f(j, k) = a_n$$
a$$
then $a_n$ a$ is $n$-perfect iconic
Is the maximum $n$-perfect iconic number the same as the maximum $n$-iconic number?
Find the maximum $n$-perfect iconic number.
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4
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edited Jul 25 2011 at 2:00 |
Story
I was bored sitting in front of my computer and using a rectangle to select icons on my screen. I could select $1$, $2$, $3$, $4$, but not $5$ icons.
(Black squares are the icons. Note that it is possible to find rectangles with $1$, $2$, $3$ ad $4$ black squares in respectively.)
So I rearranged the icons into the following arrangement:
So in this arrangement, there exists a rectangle that includes $i$ black squares, where $1 \le i \le 6$. However, on the other hand, $6$ is not the maximum. For example, we can actually achieve $7$ with this arrangement.
Problem Formulation
$a_n$ is a $n$-iconic number if there exists an arrangement function $f:\{ x \in \mathbb{Z} | 1 \le x \le n\}^2 \rightarrow \{ 0, 1\}$, such that for $1 \le i \le a_n$, there exists a quadruple $(a_i, b_i, c_i, d_i)$ (\alpha_i, \beta_i, \gamma_i, \delta_i)$ such that
$$\sum_{j = a_i}^{b_i} \alpha_i}^{\beta_i} \sum_{k = c_i}^{d_i} \gamma_i}^{\delta_i} f(j, k) = i$$
Find the maximum $n$-iconic number.
If $a_n$ is an iconic number with an arrangement function $f$ with an additional criterion
$$\sum_{j = 1}^n \sum_{k = 1}^n f(j, k) = a_n$$
then $a_n$ is $n$-perfect iconic
Is the maximum $n$-perfect iconic number the same as the maximum $n$-iconic number?
Find the maximum $n$-perfect iconic number.
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3
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edited Jul 25 2011 at 0:29 |
Story
I was bored sitting in front of my computer and using a rectangle to select icons on my screen. I could select $1$, $2$, $3$, $4$, but not $5$ icons.
(Black squares are the icons. Note that it is possible to find rectangles with $1$, $2$, $3$ ad $4$ black squares in respectively.)
So I rearranged the icons into the following arrangement:
So in this arrangement, there exists a rectangle that includes $i$ black squares, where $1 \le i \le 6$. However, on the other hand, $6$ is not the maximum. For example, we can actually achieve $7$ with this arrangement.
Problem Formulation
$a_n$ is a $n$-iconic number if there exists an arrangement function $f:\{ x \in \mathbb{Z} | 1 \le x \le n\}^2 \rightarrow \{ 0, 1\}$, such that for $1 \le i \le a_n$, there exists a quadruple $(a_i, b_i, c_i, d_i)$ such that
$$\sum_{j = a_i}^{b_i} \sum_{k = c_i}^{d_i} f(j, k) = i$$
Find the maximum $n$-iconic number.
If $a_n$ is an iconic number with an arrangement function $f$ with an additional criterion
$$\sum_{j = 1}^n \sum_{k = 1}^n f(j, k) = a_n$$
then $a_n$ is $n$-perfect iconic
Is the maximum $n$-perfect iconic number the same as the maximum $n$-iconic number.?
Find the maximum $n$-perfect iconic number.
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2
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edited Jul 25 2011 at 0:27 |
Story
I was bored sitting in front of my computer and using a rectangle to select icons on my screen. I could select $1$, $2$, $3$, $4$, but not $5$ icons.
(Black squares are the icons. Note that it is possible to find rectangles with $1$, $2$, $3$ ad $4$ black squares in respectively.)
So I rearranged the icons into the following arrangement:
So in this arrangement, there exists a rectangle that includes $i$ black squares, where $1 \le i \le 6$. However, on the other hand, $6$ is not the maximum. For example, we can actually achieve $7$ with this arrangement.
Problem Formulation
$a_n$ is a $n$-iconic number if there exists an arrangement function $f:{ f:\{ x \in \mathbb{Z} | 1 \le x \le n}^2 n\}^2 \rightarrow \{ 0, 1}$1\}$, such that for $1 \le i \le a_n$, there exists a quadruple $(a_i, b_i, c_i, d_i)$ such that
$$\sum_{j = a_i}^{b_i} \sum_{k = c_i}^{d_i} f(j, k) = i$$
Find the maximum $n$-iconic number.
If $a_n$ is an iconic number with an arrangement function $f$ with an additional criterion
$$\sum_{j = 1}^n \sum_{k = 1}^n f(j, k) = a_n$$
then $a_n$ is $n$-perfect iconic
Is the maximum $n$-perfect iconic number the same as the maximum $n$-iconic number.
Find the maximum $n$-perfect iconic number.
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1
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asked Jul 24 2011 at 21:31 |
Icon Arrangement on Desktop
Story
I was bored sitting in front of my computer and using a rectangle to select icons on my screen. I could select $1$, $2$, $3$, $4$, but not $5$ icons.
(Black squares are the icons. Note that it is possible to find rectangles with $1$, $2$, $3$ ad $4$ black squares in respectively.)
So I rearranged the icons into the following arrangement:
So in this arrangement, there exists a rectangle that includes $i$ black squares, where $1 \le i \le 6$. However, on the other hand, $6$ is not the maximum. For example, we can actually achieve $7$ with this arrangement.
Problem Formulation
$a_n$ is a $n$-iconic number if there exists an arrangement function $f:{ x \in \mathbb{Z} | 1 \le x \le n}^2 \rightarrow { 0, 1}$, such that for $1 \le i \le a_n$, there exists a quadruple $(a_i, b_i, c_i, d_i)$ such that
$$\sum_{j = a_i}^{b_i} \sum_{k = c_i}^{d_i} f(j, k) = i$$
Find the maximum $n$-iconic number.
If $a_n$ is an iconic number with an arrangement function $f$ with an additional criterion
$$\sum_{j = 1}^n \sum_{k = 1}^n f(j, k) = a_n$$
then $a_n$ is $n$-perfect iconic
Is the maximum $n$-perfect iconic number the same as the maximum $n$-iconic number.
Find the maximum $n$-perfect iconic number.
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