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Ass and Mule Problem


A classic arithmetical problem probably first posed by Euclid and investigated by various authors in the Middle Ages.

The problem is formulated as a dialogue between the two animals after which it is called. The mule say to the ass, "If you gave me one of your sacks, I would have as many as you." The ass replies, "If you gave one of your sacks, I would have twice as many as you." The question, of course, is "How many sacks do they have?" The number x of sacks of the mule and the number y of sacks of the ass are related by the identities:

x+1=y-1
(1)
2(x-1)=y+1,
(2)

which form a system of two linear equations. The solution is x=5, y=7.

A generalization of this problem has been studied by Singmaster (1999, 2002). In this modified problem, the ass says, "If you gave me a of your sacks, I would have b times as many as you," and the mule answers, "If you gave my c of your sacks, I would have d times as many as you." This has solutions

x=c+((b+1)(a+c))/(bd-1)
(3)
y=a+((d+1)(a+c))/(bd-1),
(4)

which are integers only when (bd-1)|(d+1)(a+c) and (bd-1)|(b+1)(a+c).


See also

ASS Theorem, Pons Asinorum

This entry contributed by Margherita Barile

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References

Singmaster, D. "Some Diophantine Recreations." In The Mathemagician and the Pied Puzzler: A Collection in Tribute to Martin Gardner (Ed. E. Berlekamp and T. Rodgers). Boston, MA: A K Peters, pp. 219-235, 1999.Singmaster, D. "A Variation of the Ass and Mule Problem." Crux Math. 28, 236-238, 2002.

Referenced on Wolfram|Alpha

Ass and Mule Problem

Cite this as:

Barile, Margherita. "Ass and Mule Problem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/AssandMuleProblem.html

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