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.