Problem
COM-B1-M06-P023 Operation with Three Numbers
#23
★★★★☆ Level 4 of 5
Given the triple \((1,1,1)\). In one move, one may add \(2\) to two numbers and subtract \(1\) from the third. Can \((10,10,10)\) be obtained?
Look at the sum modulo \(3\).
One move changes the sum by \(2+2-1=3\), so the sum modulo \(3\) is preserved. Initially the sum is \(3\equiv0\pmod3\), finally \(30\equiv0\pmod3\), so this invariant does not forbid it. In fact the state is reachable: apply three moves, subtracting from a different coordinate each time; all coordinates increase by \(3\). Repeating this three-move cycle three times gives \((10,10,10)\).
Useful task: not every invariant forbids; sometimes we must notice reachability.