Q1.
Take different 4-digit numbers and try carrying out these steps. Find out what happens. Check with your friends what they got. (Also complete the last column: A = ___, B = ___, C = ___)
Answer
Whatever 4-digit number you start with (as long as at least two of its digits are different), you always land on 6174 — the Kaprekar constant — and then it repeats for ever.
The chain started in the book with 6382 finishes like this:
| Round | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| A (largest) | 8632 | 6642 | 7641 | 7641 |
| B (smallest) | 2368 | 2466 | 1467 | 1467 |
| C = A − B | 6264 | 4176 | 6174 | 6174 |
So the last column is A = 7641, B = 1467, C = 7641 − 1467 = 6174. Once you reach 6174 the steps only give 6174 again and again.
Try a fresh number, say 3524:
5432 − 2345 = 3087
8730 − 3078 = 5652
6552 − 2556 = 3996
9963 − 3699 = 6264
6642 − 2466 = 4176
7641 − 1467 = 6174 ✓
8730 − 3078 = 5652
6552 − 2556 = 3996
9963 − 3699 = 6264
6642 − 2466 = 4176
7641 − 1467 = 6174 ✓
Try This: ask four friends to each pick a different 4-digit number. Everyone will end at 6174 — only the number of rounds will differ. (A number like 3333, with all four digits the same, gives 0 and is not allowed.)