Charlie Reams wrote:Great topic, enjoyed reading all this!
Harder question: if one were designing a new variant, which four large numbers (in the range 11-100) make the game hardest (i.e. fewest games solvable) and which easiest (i.e. most games solvable)?
I have an answer for you, Charlie. The easiest four numbers are:
*** Drum roll ***
59, 83, 93, 97, with 11,072,459 of the 11,905,457 games solvable, or 9,108,733 / 9,343,307 if you discount 6 small. That's 93.00% or 97.49% respectively.
The hardest four numbers are 12, 16, 18, 24, with 10,510,350 / 11,905,457 (88.28%) or 8,546,624 / 9,343,307 (91.47%).
The breakdowns by number of larges chosen is:
59, 83, 93, 97
--------------------------
6 small: 1,963,726 / 2,562,150 (76.64%)
1 large: 5,063,270 / 5,221,392 (96.97%)
2 large: 3,259,190 / 3,317,310 (98.25%)
3 large: 738,513 / 755,160 (97.80%)
4 large: 47,760 / 49,445 (96.59%)
12, 16, 18, 24
--------------------------
6 small: 1,963,726 / 2,562,150 (76.64%)
1 large: 4,704,925/ 5,221,392 (90.11%)
2 large: 3,087,519 / 3,317,310 (93.07%)
3 large: 707,920 / 755,160 (93.74%)
4 large: 46,260 / 49,445 (93.56%)
25, 50, 75, 100 performs relatively poorly, coming 2,378,830th out of the 2,555,190 possible selections - that's in the 7th percentile.
It still beats 11, 12, 13, 14, which comes 2,554,989th out of 2,555,190, with 10,621,119 / 11,905,457 (89.21%) or 8,657,393 / 9,343,307 (92.66%)
97,98,99,100 is actually not too bad, coming 1,677,651th out of 2,555,190 (35th percentile), with counts of 10,933,717 / 11,905,457 (91.84%) or 8,969,991 / 9,343,307 (96.00%)