Unlucky 13
Every natural number can be written as the sum of squares of other natural numbers. Some numbers can we expressed multiple ways.
For example:
The number 5 can be written as the sum of the squares of 2 and 1. (4 + 1 = 5). Or it can be written as the sum of 1, 1, 1, 1, and 1 (1 + 1 + 1 + 1 + 1 = 5).
Including rearranging the numbers. (5 has 3 options, 4 & 1, 1 & 5, and 1, 1, 1, 1, & 1.) How many ways can you write the number 13? Make this .
What is ?
The answer is 20.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
Thirteen can be written the following ways.
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, & 1.
2, 1, 1, 1, 1, 1, 1, 1, 1, & 1.
2, 2, 1, 1, 1, 1, & 1.
3, 1, 1, 1, & 1.
3 & 2.
The first way has only ones, so it can only be written 1 option
The second way has a 2 which can be moved moved around to occupy any of the 10 spaces, so there are 10 options.
I will come back to the 3rd option later.
The fourth way has a 1 which can be moved to occupy any of the 4 spaces, so the are 4 options.
The fifth way has a 3 which can be moved to occupy any other the 5 spaces, so there are 5 options.
The sixth ways can be written as 3 & 2, or 2 & 3, so there are 2 options.
Back to the third way. If a two occupies the first space, the second two has 6 options.
If the first two is in the second spot, the second 2 has 5 options.
Etc. 6 + 5 + 4+ 3 + 2 + 1 = 21.
Now let's add up our options.
1 + 10 + 21 + 4 + 5 + 2 = 43 = A
43 - 13 = 2 0