## Fractions as intermediate steps for some solutions

In solutions for 24 math game, we often see divisions. For example, for quadruple (1, 7, 12, 13), a soultion is (13+1)×(12/7) which requires fraction. But fraction is not necessary because an equivalent solution is (13+1)×12/7 which does not require fractions.
However for quadruple (5 7 7 11), the only solution is (5-11/7)×7 for which fraction is necessary in the intermediate steps. Fractions pose particular challenges to human brains and make solutions more difficult. So we ask, how many solutions require fractions and how many quadruples' only solution are by way of fractions.

There are 16 quadruples that their only solutions involves fractions. And they are all quite difficult to solve. Out of all 1362 solvables puzzles, they are almost all ranked in the top 200 most difficult puzzles according to our daily updated ranking of puzzle difficulties.

 Index puzzle solution(s) Rank of difficulties 131 1 3 4 6 6/(1-3/4) 2 190 1 4 5 6 6/(5/4-1) and 4/(1-5/6) 3 225 1 5 5 5 (5-1/5)×5 61 265 1 6 6 8 6/(1-6/8) 7 323 1 8 12 12 12/(12/8-1) 13 405 2 2 11 11 (2/11+2)×11 46 410 2 2 13 13 (2-2/13)×13 35 438 2 3 5 12 12/(3-5/2) 1 517 2 4 10 10 (4/10+2)×10 10 528 2 5 5 10 (5-2/10)×5 4 595 2 7 7 10 (10/7+2)×7 23 695 3 3 7 7 (3/7+3)×7 58 701 3 3 8 8 8/(3-8/3) 6 923 4 4 7 7 (4-4/7)×7 52 1083 5 5 7 11 (7-11/5)×5 20 1135 5 7 7 11 (5-11/7)×7 25

There are, in total, 103 solutions with fractions for 24 math game.
Among those, 2 each for (1 4 5 6), (1 6 9 12), (2 4 6 9) and (4 5 8 10).

 11 1 1 2 12 12/(1-1/2) 65 1 2 3 8 8/(1-2/3) 69 1 2 3 12 12/(3/2-1) 79 1 2 4 12 12/(1-2/4) 114 1 2 11 12 2/(1-11/12) 117 1 2 12 13 2/(13/12-1) 125 1 3 3 9 (3-1/3)×9 131 1 3 4 6 6/(1-3/4) 133 1 3 4 8 8/(4/3-1) 142 1 3 5 9 (5/3+1)×9 153 1 3 6 12 12/(1-3/6) 156 1 3 7 8 3/(1-7/8) 162 1 3 8 9 3/(9/8-1) 169 1 3 9 11 (11/3-1)×9 190 1 4 5 6 6/(5/4-1) and 4/(1-5/6) 199 1 4 6 7 4/(7/6-1) 200 1 4 6 8 8/(1-4/6) 204 1 4 6 12 12/(6/4-1) 215 1 4 8 12 12/(1-4/8) 223 1 4 10 12 4/(1-10/12) 225 1 5 5 5 (5-1/5)×5 242 1 5 7 10 (7/5+1)×10 247 1 5 8 9 9/(1-5/8) 259 1 5 10 12 12/(1-5/10) 265 1 6 6 8 6/(1-6/8) 275 1 6 8 8 8/(8/6-1) 276 1 6 8 9 8/(1-6/9) 277 1 6 8 10 6/(10/8-1) 282 1 6 9 10 (10/6+1)×9 283 1 6 9 12 12/(9/6-1) and   6/(1-9/12) 289 1 6 12 12 12/(1-6/12) 305 1 7 10 12 10/(1-7/12) 314 1 8 8 12 8/(1-8/12) 315 1 8 9 11 9/(11/8-1) 316 1 8 9 12 8/(12/9-1) 323 1 8 12 12 12/(12/8-1) 336 1 11 11 13 (13/11+1)×11 339 1 11 13 13 (11/13+1)×13 359 2 2 3 9 (2/3+2)×9 362 2 2 3 12 12/(2-3/2) 379 2 2 5 10 (2/5+2)×10 381 2 2 5 12 12/(5/2-2) 405 2 2 11 11 (2/11+2)×11 410 2 2 13 13 (2-2/13)×13 434 2 3 5 8 8/(2-5/3) 438 2 3 5 12 12/(3-5/2) 443 2 3 6 9 (3-2/6)×9 449 2 3 7 8 8/(7/3-2) 453 2 3 7 12 12/(7/2-3) 494 2 4 6 7 6/(2-7/4) 496 2 4 6 9 (4/6+2)×9 and   6/(9/4-2) 498 2 4 6 11 4/(2-11/6) 499 2 4 6 12 12/(2-6/4) 500 2 4 6 13 4/(13/6-2) 506 2 4 7 12 12/(4-7/2) 515 2 4 9 12 12/(9/2-4) 517 2 4 10 10 (4/10+2)×10 519 2 4 10 12 12/(10/4-2) 528 2 5 5 10 (5-2/10)×5 554 2 5 9 12 12/(5-9/2) 559 2 5 11 12 12/(11/2-5) 578 2 6 8 10 8/(2-10/6) 582 2 6 9 9 (6/9+2)×9 585 2 6 9 12 12/(2-9/6) 590 2 6 11 12 12/(6-11/2) 593 2 6 12 13 12/(13/2-6) 595 2 7 7 10 (10/7+2)×7 612 2 7 12 13 12/(7-13/2) 622 2 8 9 12 (8/12+2)×9 623 2 8 9 13 9/(2-13/8) 630 2 8 12 12 12/(2-12/8) 684 3 3 5 10 (3-3/5)×10 695 3 3 7 7 (3/7+3)×7 701 3 3 8 8 8/(3-8/3) 703 3 3 8 10 8/(10/3-3) 706 3 3 9 9 (3-3/9)×9 721 3 4 4 9 (4-4/3)×9 739 3 4 6 11 6/(3-11/4) 741 3 4 6 13 6/(13/4-3) 750 3 4 8 11 8/(4-11/3) 752 3 4 8 13 8/(13/3-4) 755 3 4 9 12 (3-4/12)×9 758 3 4 10 12 12/(3-10/4) 767 3 5 5 9 (9/5+3)×5 779 3 5 7 9 (5-7/3)×9 822 3 6 9 10 (6-10/3)×9 826 3 6 10 10 (3-6/10)×10 850 3 7 9 13 (7-13/3)×9 886 3 9 11 11 (3-9/11)×11 909 4 4 5 5 (4/5+4)×5 923 4 4 7 7 (4-4/7)×7 967 4 5 8 10 (4-8/5)×10 and   (8/10+4)×5 997 4 6 8 9 (4-8/6)×9 1057 4 9 9 12 (4-12/9)×9 1076 5 5 6 6 (6-6/5)×5 1083 5 5 7 11 (7-11/5)×5 1095 5 5 10 13 (5-13/5)×10 1125 5 6 10 12 (6-12/10)×5 1135 5 7 7 11 (5-11/7)×7

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