# Problem of the Month (May 2013)

Given some subset of the digit shapes below, what is the maximum fraction of a square they can cover? The digits can be any size and be rotated, but can not be reflected or overlap.

# ANSWERS

Solvers this month were Maurizio Morandi, Jon Palin, Joe DeVincentis, and Bryce Herdt.

Here are the best-known solutions.

One Digit
 12 / 25 5 / 18 11 / 25 11 / 25 45 / 121(Bryce Herdt) 11 / 25 12 / 25 35 / 121(Bryce Herdt) 13 / 25

Two Digits
 17 / 25 152 / 225 152 / 225 16 / 25 152 / 225 52 / 75 136 / 225 11 / 15 16 / 25 16 / 25 14 / 25 16 / 25 17 / 25 409 / 800(M. Morandi) 18 / 25 143 / 225 3 / 5 143 / 225 152 / 225 387 / 625 161 / 225 3 / 5 143 / 225 152 / 225 .6318+ 161 / 225 3 / 5 16 / 25 16 / 25 17 / 25 152 / 225 .6318+ 161 / 225 .6718+ 11 / 15 29 / 45

Three Digits
 493 / 625(M. Morandi) 493 / 625(M. Morandi) 461 / 625(M. Morandi) 493 / 625(M. Morandi) 509 / 625(M. Morandi) 492 / 625(M. Morandi) 21 / 25(M. Morandi) 4196 / 5625 4124 / 5625 4196 / 5625 1432 / 1875 4052 / 5625 1507 / 1875 483 / 625 4196 / 5625 1432 / 1875 1498 / 2025 1507 / 1875 483 / 625 508 / 625 7 / 9(M. Morandi) 1483 / 1875 1432 / 1875 1498 / 2025 1507 / 1875 1579 / 2025 1519 / 1875 1459 / 1875 477 / 625(M. Morandi) .7244+(M. Morandi) 477 / 625(M. Morandi) 493 / 625(M. Morandi) 476 / 625 509 / 625(M. Morandi) 91 / 125(M. Morandi) 477 / 625(M. Morandi) 493 / 625(M. Morandi) .8196+(Bryce Herdt) 509 / 625(M. Morandi) 449 / 625 11804 / 15625(M. Morandi) 21 / 25 477 / 625(M. Morandi) 493 / 625(M. Morandi) .8196+(Bryce Herdt) 509 / 625(M. Morandi) .8596+(Bryce Herdt) 21 / 25(M. Morandi) 508 / 625 474 / 625 3971 / 5625 4196 / 5625 4066 / 5625 4421 / 5625 474 / 625 499 / 625 3 / 4(M. Morandi) 4349 / 5625 4196 / 5625 4066 / 5625 4421 / 5625 4291 / 5625 1507 / 1875 4277 / 5625 474 / 625 499 / 625 3 / 4(M. Morandi) 492 / 625 4196 / 5625 4066 / 5625 4421 / 5625 4291 / 5625 1507 / 1875 4277 / 5625 499 / 625 3 / 4(M. Morandi) 492 / 625 7 / 9(M. Morandi) 517 / 625 29 / 36(M. Morandi) 4291 / 5625 1507 / 1875 4277 / 5625 323 / 405

Four Digits
 4613 / 5625(M. Morandi) 509 / 625(M. Morandi) 4613 / 5625(M. Morandi) 120023/140625(M. Morandi) 528 / 625(M. Morandi) 4901 / 5625(M. Morandi) 12853 / 15625(M. Morandi) 4613 / 5625(M. Morandi) 120023/140625(M. Morandi) .8729+(Bryce Herdt) 4901 / 5625(M. Morandi) 509 / 625(M. Morandi) 13197 / 15625(M. Morandi) 201 / 225(M. Morandi) 541 / 625(M. Morandi) 120023/140625(M. Morandi) .8729+(Bryce Herdt) 4901 / 5625(M. Morandi) .8788+(Bryce Herdt) 40984 / 46875(M. Morandi) 553 / 625(M. Morandi) 4622 / 5625(M. Morandi) 4471 / 5625(DeVincentis) 4571 / 5625(DeVincentis) 316781/390625(DeVincentis) 187 / 225(Jon Palin) 4622 / 5625(M. Morandi) 42248 / 50625(M. Morandi) 186 / 225(M. Morandi) 37 / 45(Jon Palin) 4571 / 5625(DeVincentis) 316781/390625(DeVincentis) 187 / 225(Jon Palin) 6016 / 7225(M. Morandi) 523 / 625(DeVincentis) 331006/390625(DeVincentis) 4622 / 5625(M. Morandi) 42248 / 50625(M. Morandi) 186 / 225(M. Morandi) 1576 / 1875(DeVincentis) 4571 / 5625(DeVincentis) 316781/390625(DeVincentis) 187 / 225(Jon Palin) 6016 / 7225(M. Morandi) 523 / 625(DeVincentis) 331006/390625(DeVincentis) 42248 / 50625(M. Morandi) 186 / 225(M. Morandi) 1576 / 1875(DeVincentis) 317 / 375(M. Morandi) 14359 / 16875(M. Morandi) 773 / 900(M. Morandi) 6016 / 7225(M. Morandi) 523 / 625(DeVincentis) 331006/390625(DeVincentis) 334727/390625(DeVincentis) 204 / 245(DeVincentis) 112094/140625(M. Morandi) 117719/140625(M. Morandi) .8685+(Bryce Herdt) 4757 / 5625(M. Morandi) 494 / 625(M. Morandi) 12941 / 15625(M. Morandi) 8 / 9(M. Morandi) 21 / 25(M. Morandi) 117719/140625(M. Morandi) .8685+(Bryce Herdt) 4757 / 5625(M. Morandi) .8772+(Bryce Herdt) 4901 / 5625(M. Morandi) 544 / 625(M. Morandi) 12709 / 15625(M. Morandi) 12941 / 15625(M. Morandi) 8 / 9(M. Morandi) 21 / 25(M. Morandi) 117719/140625(M. Morandi) .8685+(Bryce Herdt) 4757 / 5625(M. Morandi) .8772+(Bryce Herdt) 4901 / 5625(M. Morandi) .8774+(Bryce Herdt) 12941 / 15625(M. Morandi) 8 / 9(M. Morandi) 21 / 25(M. Morandi) 201 / 225(M. Morandi) 541 / 625(M. Morandi) 202 / 225(M. Morandi) .8772+(Bryce Herdt) 4901 / 5625(M. Morandi) .8774+(Bryce Herdt) 553 / 625(M. Morandi) 4541 / 5625(M. Morandi) 4622 / 5625(DeVincentis) 1287 / 1600(DeVincentis) 4703 / 5625(M. Morandi) 4471 / 5625(DeVincentis) 306781/390625(DeVincentis) 61 / 75(Jon Palin) 1179 / 1445(M. Morandi) 187 / 225(Jon Palin) 326781/390625(DeVincentis) 4622 / 5625(DeVincentis) 1287 / 1600(DeVincentis) 4703 / 5625(M. Morandi) 4706 / 5625(M. Morandi) 42977 / 50625(M. Morandi) 769 / 900(M. Morandi) 1179 / 1445(M. Morandi) 187 / 225(Jon Palin) 326781/390625(DeVincentis) 361 / 425(M. Morandi) 4622 / 5625(DeVincentis) 1287 / 1600(DeVincentis) 4703 / 5625(M. Morandi) 4706 / 5625(M. Morandi) 42977 / 50625(M. Morandi) 769 / 900(M. Morandi) 1179 / 1445(M. Morandi) 187 / 225(Jon Palin) 326781/390625(DeVincentis) 361 / 425(M. Morandi) 4706 / 5625(M. Morandi) 42977 / 50625(M. Morandi) 769 / 900(M. Morandi) 773 / 900(M. Morandi) 361 / 425(M. Morandi)

Five Digits
 632 / 735(DeVincentis) 121273/140625(M. Morandi) 4933 / 5625(DeVincentis) .8905+(DeVincentis) 125341/140625(DeVincentis) 120023/140625(M. Morandi) 4901 / 5625(DeVincentis) 212 / 225(DeVincentis) 124829/140625(DeVincentis) 4933 / 5625(DeVincentis) .8905+(DeVincentis) 125341/140625(DeVincentis) .8964+(DeVincentis) 125741/140625(DeVincentis) 1018 / 1125(DeVincentis) 4336247/5062500(DeVincentis) 4901 / 5625(DeVincentis) 212 / 225(DeVincentis) 124829/140625(DeVincentis) 4933 / 5625(DeVincentis) .8905+(DeVincentis) 125341/140625(DeVincentis) .8964+(DeVincentis) 125741/140625(DeVincentis) 1018 / 1125(DeVincentis) 4901 / 5625(DeVincentis) 212 / 225(DeVincentis) 124829 / 140625(DeVincentis) 71 / 75(DeVincentis) 41743 / 46875(DeVincentis) 214 / 225(DeVincentis) .8964+(DeVincentis) 125741/140625(DeVincentis) 1018 / 1125(DeVincentis) 1703 / 1875(DeVincentis) 42698 / 50625(DeVincentis) 43139 / 50625(DeVincentis) 6971 / 8100(DeVincentis) 43652 / 50625(DeVincentis) 4747 / 5625(DeVincentis) 2961029/3515625(DeVincentis) 4847 / 5625(DeVincentis) 4811 / 5625(DeVincentis) 4883 / 5625(DeVincentis) 3037079/3515625(DeVincentis) 43139 / 50625(DeVincentis) 6971 / 8100(DeVincentis) 43652 / 50625(DeVincentis) 8779 / 10125(M. Morandi) 14656 / 16875(DeVincentis) 7133 / 8100(DeVincentis) 4811 / 5625(DeVincentis) 4883 / 5625(DeVincentis) 3037079/3515625(DeVincentis) 3070568/3515625(DeVincentis) 43139 / 50625(DeVincentis) 6971 / 8100(DeVincentis) 43652 / 50625(DeVincentis) 8779 / 10125(M. Morandi) 14656 / 16875(DeVincentis) 7133 / 8100(DeVincentis) 4811 / 5625(DeVincentis) 4883 / 5625(DeVincentis) 3037079/3515625(DeVincentis) 3070568/3515625(DeVincentis) 8779 / 10125(M. Morandi) 14656 / 16875(DeVincentis) 7133 / 8100(DeVincentis) 2383 / 2700(DeVincentis) 3070568/3515625(DeVincentis) 1891 / 2205(DeVincentis) 632 / 735(DeVincentis) 211 / 225(DeVincentis) 4901 / 5625(DeVincentis) 121273/140625(M. Morandi) .8860+(DeVincentis) 123577/140625(DeVincentis) .8948+(DeVincentis) 124802/140625(DeVincentis) .8949+(DeVincentis) 120023/140625(M. Morandi) 211 / 225(DeVincentis) 4901 / 5625(DeVincentis) 212 / 225(DeVincentis) 15054692/17015625(DeVincentis) 71 / 75(DeVincentis) .8948+(DeVincentis) 124802/140625(DeVincentis) .8949+(DeVincentis) 1018 / 1125(DeVincentis) 4336247/5062500(DeVincentis) 211 / 225(DeVincentis) 4901 / 5625(DeVincentis) 212 / 225(DeVincentis) 15054692/17015625(DeVincentis) 71 / 75(DeVincentis) .8948+(DeVincentis) 124802/140625(DeVincentis) .8949+(DeVincentis) 1018 / 1125(DeVincentis) 212 / 225(DeVincentis) 15054692/17015625(DeVincentis) 71 / 75(DeVincentis) 214 / 225(DeVincentis) 1018 / 1125(DeVincentis) 42698 / 50625(DeVincentis) 4756 / 5625(M. Morandi) 43427 / 50625(DeVincentis) 6971 / 8100(DeVincentis) 43868 / 50625(DeVincentis) 7097 / 8100(DeVincentis) 191 / 225(DeVincentis) 193 / 225(DeVincentis) 3283 / 3825(DeVincentis) 9212 / 10625(DeVincentis) 6971 / 8100(DeVincentis) 43868 / 50625(DeVincentis) 7097 / 8100(DeVincentis) 7133 / 8100(DeVincentis) 9212 / 10625(DeVincentis) 6971 / 8100(DeVincentis) 43868 / 50625(DeVincentis) 7097 / 8100(DeVincentis) 7133 / 8100(DeVincentis) 9212 / 10625(DeVincentis) 7133 / 8100(DeVincentis)

Nine Digits
 5464 / 5625(Jon Palin)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 5/20/13.