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?

ANSWERS

The best-known solutions are shown below.

Squares in Squares

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

Triangles in Triangles

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

Squares in Rectangles

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