Track / Stage 4 / 52 of 340 #312 of 2000

Problem 312

AMC 12 late, AIME early

Combinatorics Difficulty 4.7 Find the answer

| $\left.\begin{array}{c}\text { Processes and operations } \\ \text { Examples and counterexamples. Constructions }\end{array}\right]$ |
| :---: | :---: |

On the table, there are five coins arranged in a row: the middle one is heads up, and the rest are tails up. In one operation, it is allowed to simultaneously flip exactly three adjacent coins. Is it possible, by performing such an operation several times, to achieve a state where all five coins are heads up?

Your answer

Official solution

First, let's flip the first 3 coins, then the first two will lie heads up, and the next three will lie tails up. Now let's flip the last three coins, as a result, all five coins will lie heads up.

## Answer

yes, it can.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.