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10 escapees

The way I looked at first just analysing the first ten cells, I then had a look at the prime factors of the numbers from 1-20 and drew my attention to the square numbers as they resulted in having the cells open due to an odd number of prime factors and only square numbers would have an odd number as there is a repeated root. I therefore concluded that the answer was 10, cells number 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.

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