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I have tiled my square kitchen floor with congruent square tiles. All the tiles are white, except those along the two diagonals, which are a

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I have tiled my square kitchen floor with congruent square tiles. All the tiles are white, except those along the two diagonals, which are all black (i.e. the corners are black, the center is black, and all the tiles along the diagonal in between are black). Assuming that I used 89 black tiles, how many white ones did I use?

Mar 27, 2015

#2
+101431
+11

You used 2 black tiles in every row, except in the center where there is just one.

So....there were 44 rows where 2 black tiles were present and 1 row where 1 was present.

This make 45 rows.   And since we have a square, there are the same number of "columns."

So  ... 45^2  - 89 = 1936 white tiles.......

Mar 28, 2015

#1
+17774
+11

After you subtract 1 tile for the middle, the remaining 88 tiles will be split with 1/2 of them in one diagonal and the other 1/2 in the other diagonal. Thus, each diagonal has 45 tiles (with the one in the middle being in both diagonals).

If the diagonal has an odd number of tiles, the diagonal will have as many tiles as each side has. (Check this out with 3 x 3 and 5 x 5 squares to convince yourself.

Since each side has 45 tiles, the total number of tiles will be 45 x 45 = 2025 tiles.

Of these tiles, 89 are black, so 2025 - 89 = 1936 are white.

Mar 28, 2015
#2
+101431
+11

You used 2 black tiles in every row, except in the center where there is just one.

So....there were 44 rows where 2 black tiles were present and 1 row where 1 was present.

This make 45 rows.   And since we have a square, there are the same number of "columns."

So  ... 45^2  - 89 = 1936 white tiles.......

CPhill Mar 28, 2015
#3
+101771
+1

What a great question.  I like it!

Mar 28, 2015