Problem

Source: 2016 JBMO TST- Macedonia

Tags: JMMO, Junior, 2016, Macedonia, combinatorics



We are given a $4\times4$ square, consisting of $16$ squares with side length of $1$. Every $1\times1$ square inside the square has a non-negative integer entry such that the sum of any five squares that can be covered with the figures down below (the figures can be moved and rotated) equals $5$. What is the greatest number of different numbers that can be used to cover the square?


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