Figure 1 has 1 red tile and 8 white tiles (9 total) Figure 2 has 4 red tiles and 12 white tiles (16 total) Figure 3 has 9 red tiles and 16 white tiles (25 total) Figure 4 has 16 red tiles and 20 white tiles (36 total)
Things to notice * The pattern counts for the red tiles are perfect squares (1, 4, 9, 16)
* The total number of tiles are also perfect squares (9, 16, 25, 36)
* The number of white tiles can be counted, but its much easier to use the formula W = T - R W = number of white tiles T = total number of tiles R = number of red tiles
* The pattern for the white tile counts is 8,12,16,20 so we basically add on 4 each time. The formula is W(n) = 4n+4. Plug in n = 1 and it leads to W(n) = 8 as expected. Plug in n = 2 and it leads to W = 12 etc.
The input n is the number of the figure which is a natural number. Natural numbers are {1, 2, 3, 4, ...} which are counting numbers. The function is NOT continuous. We can't plug in n = 1.5 for instance. The input does not represent the number of white tiles as that is the output.
If we plugged in n = 6, then we get W(n) = 4n+4 W(6) = 4*6+4 W(6) = 30 so figure 6 will have 30 white tiles (not 10)
Do the same for n = 8 W(n) = 4n+4 W(8) = 4*8+4 W(8) = 36 figure 8 has 36 white tiles
The sum must be larger (not equal to) the longest side. If it's exactly the same, there's no room to for the shorter two sides to angle up to form a triangle.
