Problem of the Month (April 2011)
The square of side 11 below contains squares of side 1, 2, 3, and 4. Note that 3 properties are satisfied:
1) no vertical or horizontal line passes through the interior of two squares of the same size
2) the maximum number of squares of each size are packed subject to the previous constraint
3) this is the smallest square that will hold these squares subject to the previous constraints.
For each set of positive integers, we ask for the square containing squares of those sides satisfying those 3 properties. How do the answers change if 45^{o} diagonal lines also should not pass through two squares of the same size?
ANSWERS
This month the solvers were Dave Langers, Bryce Herdt, and Evert Stenlund.
Here are the best known results for various sets of integers, sorted by the largest integer:
2
 21
4



4
 43
7
 421/42/41
8
 431
10
 4321/432
11


5
 51
10
 52
9
 53
8
 54
9
 521
11
 531
11

5321/532
13
 5421/542/541
13
 54321/5432/5431
17
 543
14 (DL)


6
 621/62/61
12 (DL)
 631/63
12 (DL)
 632
13 (DL)
 6321
14 (DL)
 64
10 (DL)
 641
14 (DL)

642
14 (DL)
 6421
15 (DL)
 643
14 (DL)
 64321/6432/6431
17 (DL)
 65
11 (DL)
 651
16 (DL)

6521/652
17 (DL)
 653
14 (DL)
 6531
17 (DL)
 65321/6532
19 (DL)
 6541/654
19 (BH)

6542
19 (DL)
 65421
22 (DL)
 65431/6543
22 (DL)
 654321/65432
23 (DL)


7
 72
13 (BH)
 721
15 (BH)
 73
13 (BH)
 731/71
14 (BH)
 7321/732
16 (BH)

74
11 (BH)
 741
15 (DL)
 742
15 (DL)
 7421
16 (DL)
 743
17 (DL)

74321/7432/7431
19 (DL)
 75
12 (BH)
 751
17 (DL)
 75321/7532/7531 7521/753/752
19 (DL)
 754
18 (DL)

7541
19 (DL)
 754321/75432/75431 7543/75421/7542
23 (DL)
 76
13 (BH)
 7621/762/761
19 (BH)

7631/763
19 (BH)
 76321/7632
20 (BH)
 7641/764
22 (BH)
 76421/7642
23 (BH)

76431/7643
26 (BH)
 76432/764321
27 (BH)
 765
19 (BH)
 7651
26 (BH)


Evert Stenlund pointed out that the diagonal conditions:
1) no vertical or horizontal line passes through the interior of two squares of the same size
2) the maximum number of squares of each size are packed subject to the previous constraint
3) no 45^{o} diagonal line passes through the interior of two squares of the same size
4) this is the smallest square that will hold these squares subject to the previous constraints.
were impossible to fulfill.
He suggested the conditions:
1) no vertical or horizontal line passes through the interior of two squares of the same size
2) the maximum number of squares of each size are packed subject to the previous constraint
3) no 45^{o} diagonal line passes through the center of a square and the interior of another square of same size
4) this is the smallest integersided square that will hold these squares subject to the previous constraints.
Here are the bestknown packings with this rule:
2
 21
8 (ES)


3
 31
12 (ES)
 321/32
14 (ES/BH)


4
 4321/432/431/43/421/42/41
16 (ES/BH)


5
 52
9 (BH)
 521
20 (ES)
 5321/532/531/53
22 (ES)

541/54/51
20 (ES)
 5421/542
22 (ES)
 54321/5432/5431/543
23 (ES)


Bryce Herdt thought the diagonal condition should be:
1) no vertical, horizontal, or 45^{o} diagonal line passes through the interior of two squares of the same size
2) the maximum number of squares of each size are packed subject to the previous constraint
3) this is the smallest square that will hold these squares subject to the previous constraints.
He, Evert Stenlund, and Dave Langers all analyzed some small cases, though I didn't care for that version of the problem.
If you can extend any of these results, please
email me.
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