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Today Puzzle #529

Puzzle No. 529鈥� Tuesday 23 July

A one armed boxer only knows how to throw a jab and an uppercut. They do not have the ability to throw consecutive uppercuts. How many different seven punch combos can they do?

Today’s #PuzzleForToday has been set by the School of Mathematics and Statistics at the University of Sheffield

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There are 34 such seven punch combos.The number of seven punch combos is the same as the number of seven letter sequences consisting of j's and u's with no two consecutive u's. We'll call a sequence of j's and u's punchable if the sequence contains no two consecutive u's and count the number of punch combinations by counting the number of punchable sequences.

Every punchable sequence starts with a j or a u, and we will count the number of punchable sequences by adding the number of punchable sequences starting with j and starting with u.

There are the same number of punchable sequences of length n as there are punchable sequences of length n+1 which start with a j (the j can be added/dropped to give the correspondence).

We can't have consecutive u's, so any punchable sequence of length n + 1 which starts with a u must have second letter j. This means the total number of punchable sequences of length n + 1 starting with a u corresponds to the number of punchable sequences of length n-1 which make up the remaining n-1 letters which follow u; j.

We can conclude that:

#(punchable length n+1) = #(punchable length n)+#(punchable length n-1):

There are two punchable sequences of length 1 and three of length 2.

It follows that the number of punchable sequence of length n is the (n+2)nd Fibonacci number. So, the number of seven punch combos the boxer can throw is 34.

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