Problem of the Month (August 2001)

Consider the following operation on a positive integer: square it and then remove all occurences of any single digit. By repeating this operation, sometimes we can eliminate all the digits of a number. For example, 36 can be completely eliminated in 8 steps as follows: 36 ⇒ 129(6) ⇒ (1)664(1) ⇒ (44)0896 ⇒ (8)02(8)16 ⇒ 4(66)5(6) ⇒ (2)0(2)5 ⇒ 2(5) ⇒ (4).

Can every number be eliminated? If not, then which numbers apparently cannot be? What numbers take the longest to be eliminated?


ANSWERS

Joseph DeVincentis conjectures that 37 is the first number that cannot be eliminated in this fashion. He notes that if this is true, then there are infinitely many numbers we can get from 37 that also cannot be eliminated.

Berend Jan van der Zwaag gave a conjectured list to those numbers less than 100 that cannot be eliminated: {37, 42, 44, 59, 61, 66, 67, 69, 72, 73, 82, 83, 86, 87, 89, 92, 93}. If anyone can eliminate any of these, let me know!

Here are two numbers that take a lot of steps to eliminate: Joseph DeVincentis gives this chain of 11 steps:

338 ⇒ 11(4)2(44) ⇒ 125(44) ⇒ 156(2)5 ⇒ (2)449(22)5 ⇒ (2)0(2)050(2)5 ⇒ 2(55)02(5) ⇒ (4)080(4) ⇒ (6)400 ⇒ 1(6)0000 ⇒ 1(00000000) ⇒ (1).

Berend Jan van der Zwaag gives this chain of 14 steps:

911 ⇒ 8(2)99(2)1 ⇒ (8)0(8)3(8)0(8)1 ⇒ (9)0601 ⇒ 36(1)20(1) ⇒ 1310(44)00 ⇒ (1)7(1)6(1)x106 ⇒ 5(77)6x1012
⇒ (3)1(3)6x1024 ⇒ 25(6)x1048 ⇒ 62(5)x1096 ⇒ 38(44)x10192 ⇒ 1(444)x10384 ⇒ 1(x10768) ⇒ (1).

I think I need to call 911 after that one! Can anyone eliminiate 911 in fewer steps? Can you find a longer chain that is needed?

Joseph Babcock showed that all powers of 10 can be eliminated in two steps.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 8/31/01.