Problem of the Month (January 2019)

What is the smallest rectangular box that can hold k each of squares of 1–n if each square must be fully supported below by squares at least as big? It appears that most of those solutions are short and long, so what are the best solutions when exactly m of the squares touch the bottom of the box?

ANSWERS

Solutions were received from Marc Lapierre.

Here are the best known results.

Stacking k Each of Squares 1–n
n \ k123
1
1
2
3
2
6
10
16
3
15
30
45
4
35
64
98
5
60
112
180
6
99
192
280
7
154
286
429
8
231
432
630
9
312
592
875
10
405
810 		1178
11
551 		1026 1548
12
713 		1342 1995
13
874 		1694 2486
14 1081 2075 ?
15 1352 2548 ?
16 1610 3060 ?
17 1875 3648 ?
18 2240 4320 ?
19 2673 5100 ?
20 3045 5848 ?
21 3498 ? ?
22 4025 ? ?
23 4620 ? ?
24 5145 ? ?
25 5852 ? ?
26 6594 ? ?
27 7332 ? ?
28 8127 ? ?
29 8976 ? ?
30 10036 ? ?
31 10935 ? ?
32 12036 ? ?
33 13338 (ML) ? ?
34 14455 (ML) ? ?
35 15635 (ML) ? ?
36 17226 (ML) ? ?
37 18492 (ML) ? ?
38 19965 (ML) ? ?
39 21594 (ML) ? ?
40 23270 (ML) ? ?
41 25132 (ML) ? ?
42 26775 (ML) ? ?
43 28676 (ML) ? ?
44 31070 (ML) ? ?
45 32964 (ML) ? ?
46 34914 (ML) ? ?
47 37500 (ML) ? ?
48 40089 (ML) ? ?
49 42245 (ML) ? ?
50 44891 (ML) ? ?
51 47891 (ML) ? ?

Stacking Squares 1–n with m Squares on Bottom
n \ m234
2
6
3
15
18
4
35
36
40
5
63
60
65
6
99
105
108
7
156
162
154
8
252
231
234
9
336
312
325
10
432
405
420
11
580
551
570
12
748
713
714
13
912
945
874
14 1200 1140 1081
15 1431 1368 1352
16 1740 1680 1650
17 2077 1980 1947
18 2464 2350 2331
19 2835 2754 2680
20 3348 3180 3128
21 3876 3640 3600
22 4480 4218 4104
23 5082 4819 4720
24 5676 5456 5330
25 6440 6090 6035
26 7238 6862 6750
27 8134 7700 ?
28 9100 8568 ?
29 10036 (ML) ? ?
30 11124 (ML) ? ?
31 12255 (ML) ? ?
32 13398 (ML) ? ?
33 14760 (ML) ? ?
34 16368 (ML) ? ?

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 2/9/19.