# Sunday Times Teaser 2666 – Multiple Calls

*by Ian Duff*

The 100 people in my office each have a two digit phone number from 00 to 99. My phone was recently rewired in error so that each numeric button now produces an incorrect digit.

In trying to phone four colleagues, I phoned double the intended number. For more than ten colleagues, using their number resulted in phoning three times the intended number.

Also if I tried to phone any colleague and then asked for their phone number, and then phoned their number and asked for their number, and so on … , it always took ten calls to phone the person I intended to call in the first place.

What numbers would I actually get if I key in the numbers 01, 23, 45, 67 and 89?

3 Comments
Leave one →

A similar solution to Brian’s, using the observation that the permutation has to be a 10-cycle, but a slightly different way of constructing the 10-cycle perms.

Any phone number xy that phones three times the intended number must have y wired to 3y mod 10.

The phone numbers that ring three times the intended number must occur within the range 4 to 33.

As there are only 3 numbers ending in each digit in this range, at least 4 digits, y, must be wired to 3y mod 10. These digits must be four of 1,2,3,4,6,7,8,9, as 0 and 5 are not wired to themselves.

This test provides a significant performance improvement.