# Rav Yonason Eibeschitz's puzzle

There is a story that the Jews were oppressed and there were harsh decrees and Rav Yonason Eibschitz was challenged by the governor to write "am yisroel chai l'olmei ad" for every Jew who lived in his town on a 2 by 4 mezuzah klaf; if he did so, the governor would stop the harsh decrees. The Rav was given an hour and indeed he managed to create a snaking puzzle with thousands of combinations of "am yisroel chai l'olmei ad".

What is the origin of this story? Supposedly it is in his sefer Tiferes Yonason but I looked and found nothing.

What math equation would be used to prove this?

See the story and puzzle.

• It seems to me that the provenance of the story and a mathematical description of its core mechanism ought to be two separate questions. And the "this" in the latter needs to be specified more clearly. – Isaac Moses Jan 9 '14 at 22:06
• Why in the world would an oppressive [gentile, presumably] governor come up with this hoop, in particular, to jump through? It sounds like maybe the story of the challenge was crafted around the neat math puzzle. – Isaac Moses Jan 9 '14 at 22:11

This site claims that it is found in the book "Sarei HaMe'ah" of Rav Yehudah Leib HaKohen Maimon on pages 131-133.

Here's how to calculate the number of times the sentence occurs in the square:

You start at the center letter ayin. To form a phrase you travel to one of the daleds in one of the four corners. In doing so, you will always travel in exactly two direction (e.g. left and up). There are four such possible combinations of two directions (left+up, right+up, left+down, right+down).

Now let's say you choose one pair of directions, say right and down. You can write any particular path from the middle ayin to the lower right daled in the corner as a sequence of the form right-down-right-down-down-...[etc];. You must take 7 steps right and 9 steps down, making 16 steps total. Therefore, the total number of such sequences is 16C7 = 11,440, where nCk is the Binomial coefficient equal to n! / (k! * (n-k)!), using factorial notation.

Since, as mentioned before, we can follow the procedure above for all four possible pairs of directions, the total number of times the phrase occurs is 11,440 * 4 = 45,760.

• Excellent answer,just what I was looking for!!! I just had one question why is it 16C7 and not 16C9? – sam Jan 10 '14 at 17:18
• Either works, because 16C7=16C9. In general, nCk=nC(n-k). – user3318 Jan 10 '14 at 17:35
• Got it ,you just chose 7,great thanks again,you don't know how long this bothered me,even if the story didn't happen its irrelevant. – sam Jan 10 '14 at 18:00