Problems & Puzzles: Puzzles

 Puzzle 306. Generalized LYM Puzzle In his beautiful puzzle's book, "Puzzles 101: A Puzzlemaster's Challenge" Nobuyuki Yoshigahara poses an extremely amusing one, the puzzle 34*, which goes like this: Select two distinct numbers a & b, from 2 to 9. Get the least number composed only by digits a or b, and divisible by both of them Example: if a= 3 & b=5, the asked least number is 3555. What is the largest of the least numbers produced for all the permitted combinations of the two numbers selected? And my logical generalization is this one: Select k distinct numbers a, b,... from 2 to 9. Get the least number composed by digits a or b or ... and divisible by all of them. Example: if k=3 and a= 3 & b=5, c=7 the asked least number is 735. What is the largest of the least numbers produced for all the permitted combinations of the k numbers selected, for all valid k? Questions (+) 1. Solve the generalized LYM puzzle. 2. Redo 1 permitting to select numbers from 1 to 9. ______ * This puzzle was named LYM (Least Yoshigahara Multiple) by Technology Review from MIT.(+) When you send your solutions please send them for each k value analyzed.

Contributions came from J. Tramu.

According to his results, the maximal LYM became for k=2, in both cases (77777779779), that is to say, the maximal LYM was for the original Nobuyuki's puzzle; the generalized version did not produce a larger solution. The next largest solution became when '1' is permitted and you use 8 different digits (1123449768)

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Here my solutions to the LYM puzzle. (Remark : all combinations including 5 and a multiple of 2 are discarded, as ending with 0). Here lym stands for the largest of the least numbers :

Question 1  : starting from '2'

k= 2
7 9   lym = 77777779779
k= 3
7 8 9   lym = 7797888
k= 4
3 7 8 9   lym = 33879888
k= 5
4 6 7 8 9   lym = 4697784
k= 6
2 3 6 7 8 9   lym = 22398768
k= 7
2 3 4 6 7 8 9   lym = 23469768

Question 2  : '1' allowed

k= 2
7 9   lym = 77777779779
k=3
7 8 9   lym = 7797888
k=4
1 7 8 9   lym = 177897888
k=5
1 3 7 8 9   lym = 13719888
k=6
2 3 6 7 8 9   lym = 22398768
k=7
2 3 4 6 7 8 9   lym = 23469768
k=8
1 2 3 4 6 7 8 9   lym = 1123449768

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