Problem of the Month(December 2013)

This month we pack squares and regular octagons in each other. What is the smallest square that contains non-overlapping octagons with sides 1-n? We can ask the same question about squares in octagons, and octagons in octagons.

Here are the best known results:
Octagons In Squares
 s = 1 + √2 s = 3 + (9/4) √2 s = 5 + (15/4) √2 s = 6 (1 + √2) s = 8 (1 + √2) s = 10 (1 + √2) s = (41 + 33√2)/3(Maurizio Morandi) s = (138 + 105√2)/8(Maurizio Morandi) s = 17(1 + √2)(Maurizio Morandi) s = (137 + 104√2)/6(Maurizio Morandi) s = 26 + 20√2(Maurizio Morandi) s = (120 + 89√2)/4(Maurizio Morandi) s = (260 + 205√2)/8 (Maurizio Morandi) s = (140 + 117√2)/4(Maurizio Morandi) s = (118 + 95√2)/3(Maurizio Morandi)

Octagons In Octagons
 s = 1 s = 3 s = 5 s = 7 s = 9 s = 11 s = 11(1 + √2)/2(Maurizio Morandi) s = (38 + 7√2)/3(Maurizio Morandi) s = (123 + 141√2)/17(Maurizio Morandi) s = 31 / √2(Maurizio Morandi) s = 25(Maurizio Morandi) s = 4 + 17√2(Maurizio Morandi) s = (115 + 294√2)/17(Maurizio Morandi) s = (359 + 164√2)/17(Maurizio Morandi) s = (151 + 83√2)/7(Maurizio Morandi)

Squares In Octagons
 s = 1 / √(2+√2)(Andrew Bayly) s = 24 – 16√2 s = 39 – 26√2 s = 13 (3√2 – 4)(Maurizio Morandi) s = 17 (3√2 – 4)(Maurizio Morandi) s = 18 – 9√2 s = 11(2 – √2)(Maurizio Morandi) s = 13(2 – √2) (Maurizio Morandi) s = 61(2 – √2)/4(Maurizio Morandi) s = 95(5 – 3√2)/7(Maurizio Morandi) s = 79(2 – √2)/4(Maurizio Morandi) s = 31(√2 – 1) (Maurizio Morandi) s = 34(√2 – 1)(Maurizio Morandi) s = 38(√2 – 1)(Maurizio Morandi) s = 102(3 – 2√2)(Maurizio Morandi)

What are the smallest octagons that can contain n non-overlapping unit squares?

Unit Squares In Octagons
 s = 1 / √(2+√2)(Andrew Bayly) s = .8766+(Maurizio Morandi) s = 6 (3 – 2√2) s = 2 / √(2+√2) s = 3 (√2 – 1) s = 7 – 4√2(Maurizio Morandi) s = 1.4369+(Maurizio Morandi) s = 1.5733+(Maurizio Morandi) s = 3 / √(2+√2) s = √(2√2)(Maurizio Morandi) s = 1.7533+(Maurizio Morandi) s = 5 (√2 – 1)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 12/1/13.