Question 190
A $3 \times 3 \times 3$ cube is divided into 27 unit-cubes. Call a *strip* any $1 \times 1 \times 3$ rectangular cuboid (block) consisting of three unit-cubes.
A positive integer is written inside each unit-cube such that any number $n$, strictly greater than 1, written in a unit-cube, is the sum of the numbers written inside three other unit-cubes, one from each of the three strips in which $n$ is situated. Prove that, regardless of the choice of the 27 numbers, there will be at least 16 among them that are smaller than or equal to 60.
A positive integer is written inside each unit-cube such that any number $n$, strictly greater than 1, written in a unit-cube, is the sum of the numbers written inside three other unit-cubes, one from each of the three strips in which $n$ is situated. Prove that, regardless of the choice of the 27 numbers, there will be at least 16 among them that are smaller than or equal to 60.