#Urgent Combinatorics help

28 messages · Page 1 of 1 (latest)

crimson star
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An ant is at a vertex of a cube. Every 10 minutes it moves to an adjacent vertex along an
edge. If N is the number of one hour journeys that end at the starting vertex, find the sum
of the squares of the digits of N .

earnest mulchBOT
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crimson star
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My approach

image.png
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I assumed that for any move that the ant is going AWAY from the blue dot ( its original spot ), it counts as +1. Any move that comes TOWARDS the dot, it counts as -1. all vertical moves count as 0.

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So

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All these blue arrows are +1's

image.png
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reverse the symbol, they are -1's.

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And

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All red arrows are counted as 0's.

image.png
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Thus by my computation we get cases where there are no 0's ( No vertical movement), 2 0's ( There can't be an odd amount since we can't get back. ) 4 0's and in the end all 0's ( just cycle about vertically )

crimson star
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but I am 22 cases short

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Plz help

grim rivet
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the hard part is finding N

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so why do you need the sum of the squares of the digits of N?

crimson star
grim rivet
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i dont get it😅

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oh wait its because N is a really big number?

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that makes sense ig

wet tangle
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this isn't cheating by any chance is it

crimson star
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Its a previous year question

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I can send u the paper

wet tangle
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ah yes ok

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so why is this "urgent"

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anyway, here's how you can do it
divide the cube into groups: startingvertex, adjacent, adjacent, oppositevertex
the number of ways to arrive at these places after 0 moves is [1,0,0,0]
after 1 move, the number of ways is [0,3,0,0]
try generalizing this to allow you to find all 60 minute paths

crimson star