Source: Mathematical Puzzles of Sam Loyd, Volume Two, edited by Martin Gardner, Dover Publications, New York, 1960, #74, #78.
There are respectively: 12 fractions equal to 1/2: 6729/13458, 6792/13584, etc.... 2 fractions equal to 1/3: 5823/17469, 5832/17496 4 fractions equal to 1/4: 3942/15768, 4392/17568, etc... 12 fractions equal to 1/5: 2697/13485, 2769/13845, etc... 3 fractions equal to 1/6: 2943/17658, 4653/17918, etc... 7 fractions equal to 1/7: 2394/16758, 2638/18459, etc... 46 fractions equal to 1/8: 3187/25496, 4589/36712 etc... 3 fractions equal to 1/9: 6381/57429, 6471/58239, etc...I asked him to send all of his solutions. They are included below.
square -> 2 squares 116795 S(70) = 4900 S(24) + 111895 S(69) 56980 S(55) = 25585 S(42) + 31395 S(45) square -> 2 triangles S(i+1) = T(i) + T(i+1), plus: 140 S(7) = 20 T(4) + 120 T(8) 3311 S(21) = 35 T(5) + 3276 T(26) 1785 S(17) = 455 T(13) + 1330 T(19) 10416 S(31) = 3276 T(26) + 7140 T(34) 40425 S(49) = 14190 T(43) + 26235 T(53) 121836 S(71) = 45760 T(64) + 76076 T(76) triangle -> 2 squares 56 T(6) = 1 S(1) + 55 S(5) 286 T(11) = 1 S(1) + 285 S(9) 10 T(3) = 5 S(2) + 5 S(2) 35 T(5) = 5 S(2) + 30 S(4) 680 T(15) = 30 S(4) + 650 S(12) 2925 T(25) = 55 S(5) + 2870 S(20) 20825 T(49) = 285 S(9) + 20540 S(39) 26235 T(53) = 650 S(12) + 25585 S(42) 7770 T(35) = 2870 S(20) + 4900 S(24) 152096 T(96) = 2870 S(20) + 149226 S(76) 125580 T(90) = 13685 S(34) + 111895 S(69) triangle -> 2 triangles 20 T(4) = 10 T(3) + 10 T(3) 680 T(15) = 120 T(8) + 560 T(14) 29260 T(55) = 1540 T(20) + 27720 T(54) 34220 T(58) = 4960 T(30) + 29260 T(55) 70300 T(74) = 10660 T(39) + 59640 T(70)
A B B/A 6381 57429 9 6471 58239 9 8361 75249 9 3187 25496 8 4589 36712 8 4591 36728 8 4689 37512 8 4691 37528 8 4769 38152 8 5237 41896 8 5371 42968 8 5789 46312 8 5791 46328 8 5839 46712 8 5892 47136 8 5916 47328 8 5921 47368 8 6479 51832 8 6741 53928 8 6789 54312 8 6791 54328 8 6839 54712 8 7123 56984 8 7312 58496 8 7364 58912 8 7416 59328 8 7421 59368 8 7894 63152 8 7941 63528 8 8174 65392 8 8179 65432 8 8394 67152 8 8419 67352 8 8439 67512 8 8932 71456 8 8942 71536 8 8953 71624 8 8954 71632 8 9156 73248 8 9158 73264 8 9182 73456 8 9316 74528 8 9321 74568 8 9352 74816 8 9416 75328 8 9421 75368 8 9523 76184 8 9531 76248 8 9541 76328 8 2394 16758 7 2637 18459 7 4527 31689 7 5274 36918 7 5418 37926 7 5976 41832 7 7614 53298 7 2943 17658 6 4653 27918 6 5697 34182 6 2697 13485 5 2769 13845 5 2937 14685 5 2967 14835 5 2973 14865 5 3297 16485 5 3729 18645 5 6297 31485 5 7629 38145 5 9237 46185 5 9627 48135 5 9723 48615 5 3942 15768 4 4392 17568 4 5796 23184 4 7956 31824 4 5823 17469 3 5832 17496 3 6729 13458 2 6792 13584 2 6927 13854 2 7269 14538 2 7293 14586 2 7329 14658 2 7692 15384 2 7923 15846 2 7932 15864 2 9267 18534 2 9273 18546 2 9327 18654 2