# Problem of the Month (August 2012)

What is the smallest square that can contain 2 or more non-overlapping squares with distinct integer areas totaling n? Can you improve the packings below, or provide other good packings for n≤100? What about equilateral triangles in equilateral triangles?

The best-known solutions are shown below.

Squares in Squares
 3s = √2 + √1 = 2.414+ 4s = √3 + √1 = 2.732+ 5s = √4 + √1 = 3 6s = √3 + √2 = 3.146+ 7s = √4 + √2 = 3.414+ 8s = √7 + √1 = 3.645+ 9s = √4 + √3 = 3.732+ 10s = √4 + √3 = 3.732+ 11s = √5 + √3 = 3.968+ 12s = √6 + √3 = 4.181+ 13s = √5 + √4 = 4.236+ 14s = √5 + √4 = 4.236+ 15s = √5 + √4 = 4.236+ 16s = √6 + √4 = 4.449+ 17s = √7 + √4 = 4.645+ 18s = √6 + √5 = 4.685+ 19s =√4+√3+√1= 4.732+ 20s = √7 + √5 = 4.881+ 21s = √8 + √5 = 5.064+ 22s = √7 + √6 = 5.095+ 23s =√4+√3+√2= 5.146+ 24s = √9 + √5 = 5.236+ 25s =√5+√3+√2= 5.382+ 26s = √9 + √6 = 5.449+ 27s =√6+√3+√2= 5.595+(Maurizio Morandi) 28s = √9 + √7 = 5.645+ 29s = √11 + √6 = 5.766+ 30s = √9 + √8 = 5.828+ 31s = √11 + √7 = 5.962+ 32s =√5+√4+√3= 5.968+ 33s =√5+√4+√3= 5.968+ 34s = √13 + √6 = 6.055+ 35s =√6+√4+√3= 6.181+ 36s = √13 + √7 = 6.251+ 37s =√7+√4+√3= 6.377+(Maurizio Morandi) 38s =√6+√5+√3= 6.417+(Maurizio Morandi) 39s = √15 + √7 = 6.518+ 40s = √13 + √9 = 6.605+ 41s =√6+√5+√4= 6.685+ 42s = √15 + √8 = 6.701+ 43s = √16 + √8 = 6.828+ 44s =√7+√5+√4= 6.881+ 45s=√10+√4+√3=6.894+ 46s=√14+√5+√1=6.977+ 47s = √14 + √11 = 7.058+ 48s =√8+√5+√4= 7.064+ 49s = √16 + √10 = 7.162+ 50s =√9+√5+√4= 7.236+ 51s = √17 + √10 = 7.285+(Joe DeVincentis) 52s =√10+√5+√4= 7.398+(Joe DeVincentis) 53s = √17 + √11 = 7.439+(Joe DeVincentis) 54s = √19 + √10 = 7.521+(Joe DeVincentis) 55s = √17 + √12 = 7.587+(Joe DeVincentis) 56s =√9+√7+√4= 7.645+(Joe DeVincentis) 57s = √18 + √12 = 7.706+(Joe DeVincentis) 58s =√10+√7+√4= 7.808+(Joe DeVincentis) 59s =√10+√6+√5= 7.847+(Joe DeVincentis) 60s = √20 + √12 = 7.936+(Joe DeVincentis) 61s = √18 + √14 = 7.984+(Joe DeVincentis) 62s = √21 + √12 = 8.046+(Joe DeVincentis) 63s = √19 + √14 = 8.100+(Joe DeVincentis) 64s = √21 + √13 = 8.188+(Joe DeVincentis) 65s = √20 + √14 = 8.213+(Joe DeVincentis) 66s = √21 + √14 = 8.324+(Andrew Bayly) 67s=√13+√4+√3+√1=8.337+(Andrew Bayly) 68s = √22 + √14 = 8.432+(Andrew Bayly) 69s =√14+√4+√3+√1= 8.473+(Andrew Bayly) 70s = √23 + √14 = 8.537+(Maurizio Morandi) 71s =√15+√4+√3+√1= 8.605+(Andrew Bayly) 72s = √24 + √14 = 8.640+(Maurizio Morandi) 73s =√14+√5+√3+√1= 8.709+(Maurizio Morandi) 74s = √25 + √14 = 8.741+(Maurizio Morandi) 75s = √26 + √14 = 8.840+(Maurizio Morandi) 76s =√13+√9+√5= 8.841+(Maurizio Morandi) 77s = √27 + √14 = 8.937+(Maurizio Morandi) 78s =√14+√9+√5= 8.977+(Maurizio Morandi) 79s = √28 + √14 = 9.033+(Maurizio Morandi) 80s = √27 + √15 = 9.069+(Maurizio Morandi) 81s =√13+√11+√5= 9.158+(Maurizio Morandi) 82s =√14+√9+√6= 9.191+(Maurizio Morandi) 83s =√16+√9+√5= 9.236+(Maurizio Morandi) 84s = √28 + √16 = 9.291+(Maurizio Morandi) 85s =√15+√9+√6= 9.322+(Maurizio Morandi) 86s = √29 + √16 = 9.385+(Maurizio Morandi) 87s =√16+√9+√6= 9.449+(Maurizio Morandi) 88s =√15+√10+√6= 9.484+(Maurizio Morandi) 89s = √31 + √16 = 9.567+(Maurizio Morandi) 90s =√16+√10+√6= 9.611+(Maurizio Morandi) 91s = √32 + √16 = 9.656+(Maurizio Morandi) 92s = √30 + √18 = 9.719+(Maurizio Morandi) 93s =√17+√9+√7= 9.768+(Maurizio Morandi) 94s = √31 + √18 = 9.810+(Maurizio Morandi) 95s =√18+√9+√7= 9.888+(Maurizio Morandi) 96s = √32 + √18 = 9.899+(Maurizio Morandi) 97s = √33 + √18 = 9.987+(Maurizio Morandi) 98s = √32 + √19 = 10.015+(Maurizio Morandi) 99s =√18+√9+√8= 10.071+(Maurizio Morandi) 100s = √33+√7+√3 = 10.122+(Maurizio Morandi)

Triangles in Triangles
 3s = √2 + √1 = 2.414+ 4s = √3 + √1 = 2.732+ 5s = √4 + √1 = 3 6s = √3 + √2 = 3.146+ 7s = √4 + √2 = 3.414+ 8s = √7 + √1 = 3.645+ 9s = √4 + √3 = 3.732+ 10s = √4 + √3 = 3.732+ 11s = √5 + √3 = 3.968+ 12s = √6 + √3 = 4.181+ 13s = √5 + √4 = 4.236+ 14s = √5 + √4 = 4.236+ 15s = √5 + √4 = 4.236+ 16s = √6 + √4 = 4.449+ 17s = √7 + √4 = 4.645+ 18s = √6 + √5 = 4.685+ 19s = 4.719+(Maurizio Morandi) 20s = √7 + √5 = 4.881+ 21s = 5.050+(Maurizio Morandi) 22s = 5.084+(Maurizio Morandi) 23s = 5.125+(Maurizio Morandi) 24s = √9 + √5 = 5.236+ 25s = 5.355+(David W. Cantrell) 26s = √9 + √6 = 5.449+ 27s = 5.563+(David W. Cantrell) 28s = 5.639+(Maurizio Morandi) 29s = √11 + √6 = 5.766+ 30s = √9 + √8 = 5.828+ 31s = 5.915+(Maurizio Morandi) 32s = 5.953+(David W. Cantrell)

Squares in Rectangles
 3A = 2 + √2 = 3.414+ 4A = 3 + √3 = 4.732+ 5A = 3 + √6 = 5.449+ 6A = 4 + 2√2 = 6.828+ 7A = 4 + 2√3 = 7.464+ 8A = 8.812+(Bryce Herdt) 9A = 6 + 2√3 = 9.464+(Bryce Herdt) 10A = 7 + √14 = 10.741+(Maurizio Morandi) 11A = 6 + √30 = 11.477+ 12A = 8 + 2√6 = 12.898+(Maurizio Morandi) 13A = 7 + √42 = 13.480+ 14A = 14.888+(Maurizio Morandi) 15A = 10 + √30 = 15.477+(Bryce Herdt) 16A = 11 + √33 = 16.744+(Bryce Herdt) 17A = 9 + 6√2 = 17.485+ 18A = 12 + 4√3 = 18.928+(Maurizio Morandi) 19A = 10 + 3√10 = 19.486+ 20A = 20.920+(Maurizio Morandi) 21A = 14 + 2√14 = 21.483+(Brian Trial) 22A = 15 + 2√15 = 22.745+(Brian Trial) 23A = 12 + 2√33 = 23.489+ 24A = 15 + 3√10 = 24.486+(Brian Trial) 25A = 13 + 2√39 = 25.489+ 26A = 26.937+(Maurizio Morandi) 27A = 18 + 3√10 = 27.486+(Maurizio Morandi) 28A = 19 + √95 = 28.746+(Maurizio Morandi) 29A = 15 + √210 = 29.491+ 30A = 20 + 2√30 = 30.954+(Maurizio Morandi) 31A = 16 + 4√15 = 31.491+ 32A = 32.949+(Maurizio Morandi)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 9/18/12.