Question 1151086
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At the end of the exercise, those and only those dogs will be sitting, whose numbers are <U>perfect squares</U>, like 1, 4, 9, 16, . . . .


The number of perfect squares from 1 to 255 is  the <U>integer part</U> of the number {{{sqrt(255)}}} = 15.97, i.e. 15.



<U>ANSWER</U>.  At the end of the exercise, 15 dogs will be sitting.
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For the full solution to the similar problem, see the lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Math-Citcle-level-problem-on-lockers-and-divisors.lesson>Math Circle level problem on lockers and divisors of integer numbers</A> 

in this site.