Shrinkable numbers
A shrinkable number is one that can be reduced to a single digit by the following process:
- Create a new number where each digit is the absolute difference between consecutive digits.
- Repeat until you have a single digit.
- No digit is allowed to be a zero at any point in the process.
For example, the 8-digit number is shrinkable: Find the smallest 5-digit shrinkable number.
The answer is 12492.
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1 solution
The number can be built up one digit at a time with a little trial-and-error.
It makes sense to begin with 1, but then to make the next digit small it should be 2. The next digit cannot be 1 or 3 or we'd get a zero on the second reduction. Make it a 4.
So let's try to make the number begin 124.
Trying any of 1 to 7 for the next digit gives 120, 121, 122, or 123 so it will fail to shrink.
If we try 8 for the fourth digit, 1248x has the problem of becoming 124(8-x) which we saw wouldn't work, since (8-x) is at most 7.
Ok then let's try 9 for the fourth digit: 1249x.
If the fifth digit is 1, 12491 becomes 1258 and then 133 so doesn't shrink.
If the fifth digit is 2, 12492 becomes 1257 then 132 then 21 then 1. It shrinks! So the solution is 1 2 4 9 2