Problem of the Month (June 2020)
What is the smallest square that n squares of side 2 and m squares of side 1 can be packed into? The answer is often the smallest integer k with 4n+m ≤ k2 and n ≤ 
k/2
2, but we are interested in the other cases. What about packing these in circles instead?
ANSWERS
Solutions were received from Maurizio Morandi and Jean Hoffmann.
The cases with yellow backgrounds have been proven.
Smallest Known Packings of n and m Squares in a Square
| n \ m | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
|
|---|
| 1
| 3
| 3
| 3
| 3
| 3
| 
3 + 1/√2 = 3.707+
| 
3.898+ (MM)
| 4
| 4
| 4
|
|---|
| 2
| 4
| 4
| 4
| 4
| 4
| 4
| 4
| 4
| 
4 + 1/√2 = 4.707+
| 
(7+√7)/2 = 4.822+ (MM)
|
|---|
| 3
| 4
| 4
| 4
| 4
| 
4 + 1/√2 = 4.707+
| 
2 + 2√2 = 4.828+
| 
4.897+
| 5
| 5
| 5
|
|---|
| 4
| 
4 + 1/√2 = 4.707+
| 5
| 5
| 5
| 5
| 5
| 5
| 5
| 5
| 
5.677+ (MM)
|
|---|
| 5
| 
4 + √2 = 5.414+
| 
4 + √2 = 5.414+
| 
4 + √2 = 5.414+
| 
4 + √2 = 5.414+
| 
5.765+ (MM)
| 
(9+√7) / 2 = 5.822+
| 
3+2√2 = 5.828+ (MM)
| 
5.885+ (MM)
| 6
| 6
|
|---|
| 6
| 6
| 6
| 6
| 6
| 6
| 6
| 6
| 6
| 6
| 6
|
|---|
| 7
| 6
| 6
| 6
| 6
| 6
| 6
| 6
| 6
| 
6 + 1/√2 = 6.707+
| 
6 + 1/√2 = 6.707+
|
|---|
| 8
| 6
| 6
| 6
| 6
| 
6 + 1/√2 = 6.707+
| 
4 + 2√2 = 6.828+
| 
4 + 2√2 = 6.828+ (MM)
| 
4 + 2√2 = 6.828+ (MM)
| 7
| 7
|
|---|
| 9
| 
6 + 1/√2 = 6.707+
| 
4 + 2√2 = 6.828+
| 7
| 7
| 7
| 7
| 7
| 7
| 7
| 7
|
|---|
| 10
| 
6 + √2 = 7.414+
| 
6 + √2 = 7.414+
| 
6 + √2 = 7.414+
| 
6 + √2 = 7.414+
| 
6 + √2 = 7.414+
| 
6 + √2 = 7.414+
| 
6 + √2 = 7.414+
| 
6 + √2 = 7.414+
| 
5+√7 = 7.645+ (MM)
| 
7/2+3√2 = 7.742+ (MM)
|
|---|
Smallest Known Packings of n and m Squares in a Circle
| n \ m | 1 | 2 | 3 | 4 | 5 | 6
|
|---|
| 1
| 
√185 / 8 = 1.700+
| 
(√73–6)/√2 = 1.798+ (JH)
| 
1.865+
| 
√2257 / 24 = 1.979+
| 
√1105 / 16 = 2.077+
| 
√4745/32 = 2.152+ (MM)
|
|---|
| 2
| 
√5 = 2.236+
| 
√5 = 2.236+
| 
√5 = 2.236+
| 
√5 = 2.236+
| 
√5785 / 32 = 2.376+
| 
5/2 = 2.500 (MM)
|
|---|
| 3
| 
5√17 / 8 = 2.576+
| 
5√17 / 8 = 2.576+
| 
√11009 / 40 = 2.623+
| 
√29 / 2 = 2.692+
| 
2.733+ (MM)
| 
2.787+ (MM)
|
|---|
| 4
| 
√34 / 2 = 2.915+
| 
√34 / 2 = 2.915+
| 
√34 / 2 = 2.915+
| 
√34 / 2 = 2.915+
| 
√34 / 2 = 2.915+
| 
√2465/16 = 3.103+ (MM)
|
|---|
| 5
| 
√10 = 3.162+
| 
√10 = 3.162+
| 
√10 = 3.162+
| 
√10 = 3.162+
| 
3.214+ (MM)
| 
13√145/48 = 3.261+ (MM)
|
|---|
If you can extend any of these results, please
e-mail me.
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