Bob was thinking of a number between 5and 15 that when squared snd added to 23 equals the next number 's squared. What is the number?

let n = the number we're thinking of...

then,

n² + 23 = (n + 1)² = n² + 2n + 1

2n = 22

n = 11

Bob was thinking of 11

check...do something !!

[121 + 23 = 144 ]

Here's a trick. If you know n^2, then just add n and n+1 and you get (n+1)^2. E.g., 5^2 = 25 and 25+5+6=36 = 6^2. This works because n^2 + n + n + 1 = n^2 + 2n + 1 = (n+1)^2,

Since 23 = 11 + 12, knowing the trick makes it obvious that you are looking for 11^2 + 11 + 12 = 11^2 + 23 = 121 + 23 = 144 = 12^2

Note that ANY odd number >=3 can be the difference of two squares. Take 87 = 43 + 44.

43^2 = 1849

44^2 = 1936

1936 - 1849 = 87