Tiled floor
This tiled floor has 109 purple tiles. How many tiles are there altogether?
Age
11 to 14
Challenge level
Problem
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Tiles along the diagonal are purple, and the other tiles are white, as shown in the diagram.
There are 109 purple tiles on the bathroom floor.
How many tiles are there altogether on the bathroom floor?
This problem is adapted from the World Mathematics Championships
Student Solutions
Answer: 3025
The sequence of all floors
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Even floors: 2 purple tiles per row (total number even)
Odd floors: 1 more purple tile than the previous floor (total number odd)
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109 purple tiles
108 = 2$\times$54 so floor length 54 has 108 purple tiles
$\therefore$ floor length 55 has 108 + 1 = 109 purple tiles
Floor length 55 has 55$^2$ = 3025 tiles
The sequence of odd-sided floors
Each odd-sided floor contains the last odd-sided floor in the middle, outlined in blue.
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4 purple tiles are added each time
Side length 2 longer each time
109 = 1 + 4$\times$?
= 1 + 108
= 1 + 4$\times$27
So side length = 1 + 2$\times$27 = 55
$\therefore$ total number of tiles = 55$^2$ = 3025
Finding the number of rows
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Except for the middle row which has 1 purple tile (on odd-sided floors)
109 tiles = 1 purple tile in the middle
+ 108 purple tiles in pairs
= 1 purple tile in the middle
+ 54 pairs on 54 rows
$\therefore$ there are 1 + 54 = 55 rows
So there are 55$^2$ = 3025 tiles