# 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 45o diagonal lines also should not pass through two squares of the same size?

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:

 2 214
 3 316 325 3217
 4 437 421/42/418 43110 4321/43211
 5 5110 529 538 549 52111 53111
 5321/53213 5421/542/54113 54321/5432/543117 54314 (DL)
 6 621/62/6112 (DL) 631/6312 (DL) 63213 (DL) 632114 (DL) 6410 (DL) 64114 (DL)
 64214 (DL) 642115 (DL) 64314 (DL) 64321/6432/643117 (DL) 6511 (DL) 65116 (DL)
 6521/65217 (DL) 65314 (DL) 653117 (DL) 65321/653219 (DL) 6541/65419 (BH)
 654219 (DL) 6542122 (DL) 65431/654322 (DL) 654321/6543223 (DL)
 7 7213 (BH) 72115 (BH) 7313 (BH) 731/7114 (BH) 7321/73216 (BH)
 7411 (BH) 74115 (DL) 74215 (DL) 742116 (DL) 74317 (DL)
 74321/7432/743119 (DL) 7512 (BH) 75117 (DL) 75321/7532/75317521/753/75219 (DL) 75418 (DL)
 754119 (DL) 754321/75432/754317543/75421/754223 (DL) 7613 (BH) 7621/762/76119 (BH)
 7631/76319 (BH) 76321/763220 (BH) 7641/76422 (BH) 76421/764223 (BH)
 76431/764326 (BH) 76432/76432127 (BH) 76519 (BH) 765126 (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 45o 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 45o diagonal line passes through the center of a square and the interior of another square of same size
4) this is the smallest integer-sided square that will hold these squares subject to the previous constraints.

Here are the best-known packings with this rule:

 2 218 (ES)
 3 3112 (ES) 321/3214 (ES/BH)
 4 4321/432/431/43/421/42/4116 (ES/BH)
 5 529 (BH) 52120 (ES) 5321/532/531/5322 (ES)
 541/54/5120 (ES) 5421/54222 (ES) 54321/5432/5431/54323 (ES)

Bryce Herdt thought the diagonal condition should be:

1) no vertical, horizontal, or 45o 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 e-mail me. Click here to go back to Math Magic. Last updated 4/17/11.