1. Mysterious number 6174

2. Indian mathematician born in 1905 in Dahanu near Bombai. His work related to the theory of numbers, fractions, and periodic magical squares. Despite having no formal postgraduate training and working as a schoolteacher He published extensively and became well known in recreational mathematics circles He is also the discoverer of the constant Kaprekara. He died in 1986.

3. PROCESS: choose a four digit number where the digits are not all the same rearrange the digits to get the largest and smallest numbers these digits can make subtract the smallest number from the largest to get a new number Carry on repeating the operation for each new number.

4. Choose 2005 The subtractions are: 5200 - 0025 = 5175 7551 - 1557 = 5994 9954 - 4599 = 5355 5553 - 3555 = 1998 9981 - 1899 = 8082 8820 - 0288 = 8532 8532 - 2358 = 6174 7641 - 1467 = 6174 When we reach 6174 the operation repeats itself, returning 6174 every time. For 2005 the process reached 6174 in seven steps.

5. Choose 1789 The subtractions are: 9871 - 1789 = 8082 8820 - 0288 = 8532 8532 - 2358 = 6174 We reached 6174 again! It's marvellous, isn't it? For 1789 the process reached 6174 in three steps.

6. Suppose that 9 ≥ a ≥ b ≥ c ≥ d ≥ 0 where a, b, c, d are not all the same digit, the maximum number is abcd and the minimum is dcba

8. which gives the relations D = 10 + d - a (as a > d) C = 10 + c - 1 - b = 9 + c - b (as b > c - 1) B = b - 1 - c (as b > c) A = a - d for those numbers where a>b>c>d. So we can find the kernels of Kaprekar's operation by considering all the possible combinations of {a, b, c, d} The solution to the simultaneous equations is a=7, b=6, c=4 and d=1. That is ABCD = 6174

9. applying Kaprekar's operation to the three digit number 753 gives: 753 - 357 = 396 963 - 369 = 594 954 - 459 = 495 954 - 459 = 495 The number 495 is the unique kernel for the operation on three digit numbers.

10. We have seen that four and three digit numbers reach a unique kernel, but how about other numbers ? say 28:82 - 28 = 54 54 - 45 = 9 90 - 09 = 81 81 - 18 = 63 63 - 36 = 27 72 - 27 = 45 54 - 45 = 9 It doesn't take long to check that all two digit numbers will reach the loop 9→81→63→27→45→9. Unlike for three and four digit numbers, there is no unique kernel for two digit numbers.

11. To answer this we would need to use a similar process as before: check the 120 combinations of {a,b,c,d,e} for ABCDE such that 9 ≥ a ≥ b ≥ c ≥ d ≥ e ≥ 0 and abcde - edcba = ABCDE. Thankfully the calculations have already been done by a computer, and it is known that there is no kernel for Kaprekar's operation on five digit numbers. But all five digit numbers do reach one of the following three loops: 71973→83952→74943→62964→71973 75933→63954→61974→82962→75933 59994→53955→59994

12. DigitsKernel 2None 3495 46174 5None 6549945, 631764 7None 863317664, 97508421 9554999445, 864197532 10 6333176664, 9753086421, 9975084201 It appears that Kaprekar's operation takes every number to a unique kernel only for three and four digit numbers.

15. 2 * 142857 = 285714 3 * 142857 = 428571 4 * 142857 = 571428 5 * 142857 = 714285 6 * 142857 = 857142 Each time we get to a number, which consists of the same six digits which occurs in the number of 142 857, and in addition arranged in the same sequence, but starting from different numbers each time. 7 * 142857 = 999999 999,999 divided by 9 is 111111 7 * 285714 = 1999998 1999998 divided by 9 is 222222 7 * 428751 = 2999997 2999997 divided by 9 is 333333 7 * 571428 = 3999996 3999996 divided by 9 is 444444 7 * 714285 = 4999995 4999995 divided by 9 is 555555 7 * 857142 = 5999994 5999994 divided by 9 is 666666

16. 1 x 1 = 1 11 x 11 = 121 111 x 111 = 12321 1111 x 1111 = 1234321 11111 x 11111 = 123454321 111111 x 111111 = 12345654321 1111111 x 1111111 = 1234567654321 11111111 x 11111111 = 123456787654321 111111111 x 111111111 = 12345678987654321

18. 1 x 9 + 2 = 11 12 x 9 + 3 = 111 123 x 9 + 4 = 1111 1234 x 9 + 5 = 11111 12345 x 9 + 6 = 111111 123456 x 9 + 7 = 1111111 1234567 x 9 + 8 = 11111111 12345678 x 9 + 9 = 111111111 123456789 x 9 + 10 = 1111111111

19. 1 x 8 + 1 = 9 12 x 8 + 2 = 98 123 x 8 + 3 = 987 1234 x 8 + 4 = 9876 12345 x 8 + 5 = 98765 123456 x 8 + 6 = 987654 1234567 x 8 + 7 = 9876543 12345678 x 8 + 8 = 98765432 123456789 x 8 + 9 = 987654321