KA-0012Casework
5 points, difficulty 3 of 3Level 3–4about 105sHow many two-digit numbers have digits that add up to 8?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
List by the first digit, starting at 1.
2The strategy
The first digit decides everything, because the second digit is then forced. Run the first digit from 1 upward and stop when the second digit would have to be negative.
3The full solution
First digit 1 gives 17, 2 gives 26, 3 gives 35, 4 gives 44, 5 gives 53, 6 gives 62, 7 gives 71, 8 gives 80. First digit 9 would need a second digit of -1. That is 8 numbers.
Solution
The reliable way
Let the first digit run from 1 upwards; the second digit is then forced to be 8 minus the first. First digit 1 through 8 gives 17, 26, 35, 44, 53, 62, 71, 80 - all valid. First digit 9 would need a second digit of minus one, which is impossible, and first digit 0 is not allowed in a two-digit number. So there are 8. The transferable idea: choose the variable that forces everything else, then list in order.
The elegant way
The first digit can be anything from 1 to 8, and each choice determines the number completely, so there are exactly 8.
Why this is on the test: It rewards a listing order that cannot skip a case, and it punishes forgetting that 80 works while 08 does not.