## Brainteaser 1757: Enigma variations

**From The Sunday Times, 19th May 1996** [link]

I recently joined

Enigma, the international club for puzzle-solvers. When I received my membership card I noticed that my membership number was a six-figure number and that its digits could be rearranged to make three consecutive two-figure numbers.I also noticed that its six digits could be rearranged to form two consecutive three-figure numbers.

Furthermore, no one with a lower six-figure membership number could make the same claims.

What is my membership number?

This puzzle was included in the book *Brainteasers* (2002, edited by Victor Bryant). The puzzle text above is taken from the book.

[teaser1757]

## Jim Randell 8:51 am

on19 November 2019 Permalink |We can examine sequences of 3 consecutive 2-digit numbers (there are only 88 of them), and look for ones whose digits can be rearranged to form 2 consecutive 3-digit numbers. Each of these produces a collection of 6-digit membership numbers, and we can choose the smallest of these.

This Python program runs in 135ms.

Run:[ @repl.it ]Solution:Your membership number is: 102222.The six digits of 102222 can be rearranged as (20, 21, 22) and also (220, 221).

The largest such number is 999987:

There are 19 viable sequences of 3 consecutive 2-digit numbers, which give a total of 1195 candidate membership numbers.

If leading zeros were allowed (which would not be uncommon for membership numbers) the lowest viable membership number would be 012222 (and there would be 1350 candidate numbers).

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