For which positive integers N does there exist a square that can be completely tiled with N rectangles of integer sides whose areas or perimeters are precisely 1, 2, 3, ..., N?
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asked Mar 5, 2019 at 0:42
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2$\begingroup$ This question seem ambiguous; do you mean tiled by rectangles $R_1, R_2, ..., R_N$ such that for each $i$, either $P(R_i) = i$ or $A(R_i) = i$? In other words, does the choice of "area or perimeter" for one rectangle constrain the choice for other rectangles? Also, do you require that the rectangles have integer side lengths? $\endgroup$– user44191Commented Mar 5, 2019 at 1:13
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$\begingroup$ @user44191: yes. $\endgroup$– Bernardo Recamán SantosCommented Mar 5, 2019 at 1:15
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1$\begingroup$ Please expand the question, because your question, the questions in a comment and your answer in another comment are equally confusing, at least for me. If you are asking two questions, one about perimeters and another about areas, it is better to separate them. Just ask about areas, then add - the same question with areas replaced by perimeters. $\endgroup$– მამუკა ჯიბლაძეCommented Mar 5, 2019 at 8:42
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$\begingroup$ You came back to edit, Bernardo, but you failed to edit in any clarification on the rectangles having integer sides. $\endgroup$– Gerry MyersonCommented Mar 5, 2019 at 11:55
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1$\begingroup$ @GerryMyerson: Done. $\endgroup$– Bernardo Recamán SantosCommented Mar 5, 2019 at 12:00
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3 Answers
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Here are solutions for $N=10$ in a square of side $7$:
Here is a solution for $N=52$ in a square of side $47$:
At user44191's suggestion, here is the list of areas and perimeters of the rectangles used in $N=52$:
$$ \small \begin{array}{rlrlrl} n & \textrm{Area or Perimeter} & n & \textrm{Area or Perimeter} & n & \textrm{Area or Perimeter}\\ 1 & 1\times 1 & 19 & 19\times 1 & 37 & 37\times 1 \\ 2 & 2\times 1 & 20 & 5+5+5+5 & 38 & 10+10+9+9 \\ 3 & 3\times 1 & 21 & 7\times 3 & 39 & 13\times 3 \\ 4 & 1+1+1+1 & 22 & 6+6+5+5 & 40 & 10+10+10+10\\ 5 & 5\times 1 & 23 & 23\times 1 & 41 & 41\times 1 \\ 6 & 2+2+1+1 & 24 & 6+6+6+6 & 42 & 11+11+10+10\\ 7 & 7\times 1 & 25 & 5\times 5 & 43 & 43\times 1 \\ 8 & 3+3+1+1 & 26 & 7+7+6+6 & 44 & 11+11+11+11\\ 9 & 3\times 3 & 27 & 9\times 3 & 45 & 9\times 5 \\ 10 & 4+4+1+1 & 28 & 7+7+7+7 & 46 & 12+12+11+11\\ 11 & 11\times 1 & 29 & 29\times 1 & 47 & 47\times 1 \\ 12 & 5+5+1+1 & 30 & 8+8+7+7 & 48 & 12+12+12+12\\ 13 & 13\times 1 & 31 & 31\times 1 & 49 & 7\times 7 \\ 14 & 6+6+1+1 & 32 & 8+8+8+8 & 50 & 13+13+12+12\\ 15 & 15\times 1 & 33 & 11\times 3 & 51 & 17\times 3 \\ 16 & 5+5+3+3 & 34 & 9+9+8+8 & 52 & 13+13+13+13\\ 17 & 17\times 1 & 35 & 7\times 5 \\ 18 & 18\times 1 & 36 & 9+9+9+9 \\ \end{array} $$
answered Mar 6, 2019 at 21:48
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1$\begingroup$ Wow. I didn't think there would be an example this small. Gerhard "Much Less Four Of Them" Paseman, 2019.03.06. $\endgroup$ Commented Mar 6, 2019 at 22:37
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2$\begingroup$ Yes, @მამუკაჯიბლაძე, there is at least one solution for $N=52$. $\endgroup$ Commented Mar 9, 2019 at 2:38
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3$\begingroup$ @JosephO'Rourke It is OK, one is counted for area and another for perimeter! $\endgroup$ Commented Mar 9, 2019 at 3:47
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Here's a nice solution. N=71 into a square of 71. There is no "spare area", every rectangle is the maximum area. That means all even rectangles from 18 up use the most convex perimeter and below 18 no perimeters are used. This is the only N for which this is true except for the degenerate case of N=1.
There is still some latitude in that any non-prime odd rectangles, and even rectangles less than 18 have more than one shape available.
List of rectangles used. 'A' means area, 'P' means perimeter. Rectangle dimensions x*y are shown. To get perimeter from x,y use 2(x+y).
A 1 1x 1
A 2 1x 2
A 3 1x 3
A 4 1x 4
A 5 1x 5
A 6 2x 3
A 7 1x 7
A 8 2x 4
A 9 3x 3
A10 1x10
A11 1x11
A12 1x12
A13 1x13
A14 1x14
A15 1x15
A16 4x 4
A17 1x17
P18 4x 5
A19 1x19
P20 5x 5
A21 3x 7
P22 5x 6
A23 1x23
P24 6x 6
A25 5x 5
P26 6x 7
A27 3x 9
P28 7x 7
A29 1x29
P30 7x 8
A31 1x31
P32 8x 8
A33 1x33
P34 8x 9
A35 5x 7
P36 9x 9
A37 1x37
P38 9x10
A39 3x13
P40 10x10
A41 1x41
P42 10x11
A43 1x43
P44 11x11
A45 5x 9
P46 11x12
A47 1x47
P48 12x12
A49 7x 7
P50 12x13
A51 3x17
P52 13x13
A53 1x53
P54 13x14
A55 5x11
P56 14x14
A57 3x19
P58 14x15
A59 1x59
P60 15x15
A61 1x61
P62 15x16
A63 7x 9
P64 16x16
A65 5x13
P66 16x17
A67 1x67
P68 17x17
A69 3x23
P70 17x18
A71 1x71
List of areas for N up to 125.
1. N
2. Sum of 1 to N, ie what you get if all rectangles are of area N
3. Largest prime less than N, ie minimum size of square
4. How much extra area you get by using the largest area perimeters for each even N of 18 and higher
5. Sum of all 'extra' areas from 18 to N
6. Area of minimum square
7. Column 1 (base area) plus column 4 (extra area) minus area of minimum square. If this is zero or positive we could make a square.
1 1 1 0 0 1 0
2 3 2 0 0 4 -1
3 6 3 0 0 9 -3
4 10 3 0 0 9 1
5 15 5 0 0 25 -10
6 21 5 0 0 25 -4
7 28 7 0 0 49 -21
8 36 7 0 0 49 -13
9 45 7 0 0 49 -4
10 55 7 0 0 49 6
11 66 11 0 0 121 -55
12 78 11 0 0 121 -43
13 91 13 0 0 169 -78
14 105 13 0 0 169 -64
15 120 13 0 0 169 -49
16 136 13 0 0 169 -33
17 153 17 0 0 289 -136
18 171 17 2 2 289 -116
19 190 19 0 2 361 -169
20 210 19 5 7 361 -144
21 231 19 0 7 361 -123
22 253 19 8 15 361 -93
23 276 23 0 15 529 -238
24 300 23 12 27 529 -202
25 325 23 0 27 529 -177
26 351 23 16 43 529 -135
27 378 23 0 43 529 -108
28 406 23 21 64 529 -59
29 435 29 0 64 841 -342
30 465 29 26 90 841 -286
31 496 31 0 90 961 -375
32 528 31 32 122 961 -311
33 561 31 0 122 961 -278
34 595 31 38 160 961 -206
35 630 31 0 160 961 -171
36 666 31 45 205 961 -90
37 703 37 0 205 1369 -461
38 741 37 52 257 1369 -371
39 780 37 0 257 1369 -332
40 820 37 60 317 1369 -232
41 861 41 0 317 1681 -503
42 903 41 68 385 1681 -393
43 946 43 0 385 1849 -518
44 990 43 77 462 1849 -397
45 1035 43 0 462 1849 -352
46 1081 43 86 548 1849 -220
47 1128 47 0 548 2209 -533
48 1176 47 96 644 2209 -389
49 1225 47 0 644 2209 -340
50 1275 47 106 750 2209 -184
51 1326 47 0 750 2209 -133
52 1378 47 117 867 2209 36
53 1431 53 0 867 2809 -511
54 1485 53 128 995 2809 -329
55 1540 53 0 995 2809 -274
56 1596 53 140 1135 2809 -78
57 1653 53 0 1135 2809 -21
58 1711 53 152 1287 2809 189
59 1770 59 0 1287 3481 -424
60 1830 59 165 1452 3481 -199
61 1891 61 0 1452 3721 -378
62 1953 61 178 1630 3721 -138
63 2016 61 0 1630 3721 -75
64 2080 61 192 1822 3721 181
65 2145 61 0 1822 3721 246
66 2211 61 206 2028 3721 518
67 2278 67 0 2028 4489 -183
68 2346 67 221 2249 4489 106
69 2415 67 0 2249 4489 175
70 2485 67 236 2485 4489 481
71 2556 71 0 2485 5041 0
72 2628 71 252 2737 5041 324
73 2701 73 0 2737 5329 109
74 2775 73 268 3005 5329 451
75 2850 73 0 3005 5329 526
76 2926 73 285 3290 5329 887
77 3003 73 0 3290 5329 964
78 3081 73 302 3592 5329 1344
79 3160 79 0 3592 6241 511
80 3240 79 320 3912 6241 911
81 3321 79 0 3912 6241 992
82 3403 79 338 4250 6241 1412
83 3486 83 0 4250 6889 847
84 3570 83 357 4607 6889 1288
85 3655 83 0 4607 6889 1373
86 3741 83 376 4983 6889 1835
87 3828 83 0 4983 6889 1922
88 3916 83 396 5379 6889 2406
89 4005 89 0 5379 7921 1463
90 4095 89 416 5795 7921 1969
91 4186 91 0 5795 8281 1700
92 4278 91 437 6232 8281 2229
93 4371 91 0 6232 8281 2322
94 4465 91 458 6690 8281 2874
95 4560 91 0 6690 8281 2969
96 4656 91 480 7170 8281 3545
97 4753 97 0 7170 9409 2514
98 4851 97 502 7672 9409 3114
99 4950 97 0 7672 9409 3213
100 5050 97 525 8197 9409 3838
101 5151 101 0 8197 10201 3147
102 5253 101 548 8745 10201 3797
103 5356 103 0 8745 10609 3492
104 5460 103 572 9317 10609 4168
105 5565 103 0 9317 10609 4273
106 5671 103 596 9913 10609 4975
107 5778 107 0 9913 11449 4242
108 5886 107 621 10534 11449 4971
109 5995 109 0 10534 11881 4648
110 6105 109 646 11180 11881 5404
111 6216 109 0 11180 11881 5515
112 6328 109 672 11852 11881 6299
113 6441 113 0 11852 12769 5524
114 6555 113 698 12550 12769 6336
115 6670 113 0 12550 12769 6451
116 6786 113 725 13275 12769 7292
117 6903 113 0 13275 12769 7409
118 7021 113 752 14027 12769 8279
119 7140 119 0 14027 14161 7006
120 7260 119 780 14807 14161 7906
121 7381 119 0 14807 14161 8027
122 7503 119 808 15615 14161 8957
123 7626 119 0 15615 14161 9080
124 7750 119 837 16452 14161 10041
125 7875 119 0 16452 14161 10166
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$\begingroup$ Great contribution. This seems to support the claim that such tilings are possible for all sufficiently large N. $\endgroup$ Commented Jun 30, 2019 at 20:55
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I'll ignore the perimeters and just look at the question where the areas are meant to be $1,2,\dots,N$. Then the total area is $N(N+1)/2$, so the side of the square is less than $(N+1)/\sqrt2$. But for $N$ sufficiently large, it's guaranteed that there's a prime $p$ between $(N+1)/\sqrt2$ and $N$, so you have to use a rectangle of area $p$, and the only such rectangle with integer sides is the $1\times p$ rectangle, and that won't fit in the square (unless you tilt it, and it seems highly unlikely that there would be a way to tile a square with tilted rectangles).
answered Mar 5, 2019 at 4:54
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$\begingroup$ However, integer sides were not specified. I have eight rectangles of sides 6 by j/6 which tile a square of area 36. Gerhard "There Are Other Solutions Too" Paseman, 2019.03.04. $\endgroup$ Commented Mar 5, 2019 at 6:09
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$\begingroup$ @Gerhard, in the comments on the question, user44191 asked, "do you require that the rectangles have integer side lengths?" and OP replied, "yes". $\endgroup$ Commented Mar 5, 2019 at 6:15
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$\begingroup$ In which case you don't have to ignore perimeters, and your solution still applies. Gerhard "What To Do With Primes" Paseman, 2019.03.05. $\endgroup$ Commented Mar 5, 2019 at 15:35
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$\begingroup$ @GerhardPaseman Why does it still apply? Gerry Myerson's solution only answered the question by calculating total area, which doesn't work in the more general case where some are perimeters instead of areas. $\endgroup$ Commented Mar 5, 2019 at 20:06
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1$\begingroup$ A puzzle based on this problem is now at: puzzling.stackexchange.com/questions/85475/…. $\endgroup$ Commented Jun 26, 2019 at 1:57