Central Sum
Can you find numbers between 100 and 999 that have a middle digit equal to the sum of the other two digits?
Age
11 to 14
Challenge level
Problem
In how many whole numbers between 100 and 999 is the middle digit equal to the sum of the other two digits?
Student Solutions
Answer: 45 numbers
Considering different middle digits
| Middle digit | Possible numbers |
| 1 | 110 |
| 2 |
121 220 |
|
3
|
132 231 330 |
|
4
|
143 242 341 440 |
There is a pattern!
There will be 5 numbers with middle digit 5, 6 with middle digit 6, ... 9 with middle digit 9
So there are a total of 1+2+3+4+5+6+7+8+9 = 45 such numbers.
Considering different first digits
| In the 100s | In the 200s | In the 300s | In the 400s | ... | In the 900s |
|
110
121 132
143 154 165 176 187 198 |
220 231 242 253 264 275 286 297 |
330 341 352 363 374 385 396 |
440 451 462 473 484 495 |
The list is one shorter each time! |
990
|
So there are a total of 9+8+7+6+5+4+3+2+1 = 45 such numbers.