Problem 5

Two squirrels, Bushy and Jumpy, have collected

2021

walnuts for the winter. Jumpy numbers the walnuts from

1

through

2021

and digs

2021

little holes in a circular pattern in the ground around their favourite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention to the numbering. Unhappy, Jumpy decides to reorder the walnuts by performing a sequence of

2021

moves. In the

k

-th move, Jumpy swaps the positions of the two walnuts adjacent to walnut

k

. Prove that there exists a value of

k

such that, on the

k

-th move, Jumpy swaps some walnuts

a

and

b

with

a<k<b

Step 4 of 5: Show even blocks freeze at length

2

or split unevenly

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In plain words

A block of exactly two black walnuts can never be broken, because turning either one red would require its outside neighbour and its partner to match colour, which they do not.

L\ \text{even},\ L\ge4:\quad L-1=i+(L-1-i)

Detailed analysis

By Step 2, a walnut can only turn red when both its neighbours already match in colour. So a black block of length exactly $2$ can never lose a walnut, since each of its two walnuts has one black partner and one red outside neighbour — mismatched colours. A black block of even length $L\ge4$ , when an interior walnut turns red (necessarily with two black neighbours), splits into two sub-blocks of lengths $i$ and $L-1-i$ with $i+(L-1-i)=L-1$ odd, so one sub-block is even and the other odd.