Hop, Hop, Hop

Number Theory Level 3

Grey and Brownie--the rabbits--are initially sitting on stools #2 and #6, respectively. As shown in the diagram below left, both rabbits can only hop counter-clockwise, with Grey moving 2 stools at a time and Brownie moving 3 stools at a time.

Now, they want to switch their seats as in the diagram on the right, with minimal hops for each. Which rabbit has to hop more times?

Grey Brownie The number of hops needed is the same for both

1 solution

Worranat Pakornrat
May 19, 2017

Let $n$ be number of Brownie's hops and $m$ be Grey's ones.

Brownie would need $4$ steps to reach the destination.

Thus, $4\equiv 3n \pmod{7}$ . Then $n=6$ , for $3\cdot 6 \equiv 18 \equiv 4 \pmod{7}$ .

Similarly, Grey will need $3$ steps.

Then $3\equiv 2m \pmod{7}$ . Then $m=5$ because $2\cdot 5 \equiv 10 \equiv 3 \pmod{7}$ .

Finally, Brownie hopped $6$ times, and Grey $5$ times. So Brownie hops more than Grey.

Hop, Hop, Hop

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