A probability problem by Áron Bán-Szabó

Probability Level 4

What is the minimum value of n n , such that we are able to place one each of 1 × 1 , 1 × 2 , 1 × 3 , … , 1 × n 1\times 1, 1\times 2, 1\times 3, \dots, 1\times n tiles inside a 10 × 10 10\times 10 board in such a manner that we do not have sufficient space to place an 1 × ( n + 1 ) 1\times (n+1) tile inside the same board?

Details and Assumptions:

  • The tiles are not allowed to stick out of the board or overlap.

  • We get to choose the placement of the tiles.

6 4 8 9 7 10 5

Áron Bán-Szabó by Áron Bán-Szabó , 17, Hungary

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1 solution

Áron Bán-Szabó
Aug 27, 2017

The figure above shows that n ≤ 6 n\leq 6 .

Now I will prove that 6 satisfies the conditions.

In the figure above 16 1 × 6 1\times 6 rectangles are highlighted. So we need to cover at least one square from each 1 × 6 1\times 6 rectangle. These together only contain 1 + 2 + 3 + 4 + 5 < 16 1+2+3+4+5<16 squares, so at least one 1 × 6 1\times 6 rectangle will be free.

Therefore the answer is 6 \boxed{6} .

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