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smallest no of 5x1 tiles which must be placed on a 31x5 rectangle

Source: Austrian Polish 2003 APMC

April 25, 2020
geometryrectangle

Problem Statement

What is the smallest number of 5×15\times 1 tiles which must be placed on a 31×531\times 5 rectangle (each covering exactly 55 unit squares) so that no further tiles can be placed? How many different ways are there of placing the minimal number (so that further tiles are blocked)? What are the answers for a 52×552\times 5 rectangle?
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