Magic Numbers

[Michael Wallace]

Having just watched this reincarnation of Talking Telephone Numbers for the first time, it got me thinking of a puzzle.

For those who are unfamiliar, Magic Numbers is a (pretty terrible) game show where they generate six (single digit) numbers, and then if two of these feature in the last six digits of your mobile or landline phone number then you can call up to enter the draw to play the prize game at the end of the programme. For instance, if your phone number ended 123456 and the show's six numbers were 5,6, 7, 8, 9 and 0 you would win (you've 'matched' 5 and 6), but if your phone number was 123455 you wouldn't (you've only 'matched' 5)

Ignoring dialling codes, there are one million different possible phone numbers (000000 to 999999). If the six numbers they generate on the show are all different, how many of these million phone numbers will match at least two of the show numbers and be 'winners'?

(For clarity, on the show they say that "if two or more of these numbers are in the last six digits of your phone number", which I am taking to mean that repeated numbers are irrelevant. Therefore it is impossible to win if all six digits of your phone number are the same.)

[Howard Somerset]

I saw a bit of this programme a week ago, and I did a quick calulation that if the last six numbers on a viewer's phone were all different, you'd have about a 95% chance of matching two of the selected numbers, whatever they are. You're allowed to use landlines or mobiles, and as my wife and I happened to be together at the time, that would have been three phones we could have used, so the prob of getting at least one of these phones to match is almost 99.99%. In fact, not surprisingly, all three of our phone numbers matched.

I realise that this answers a different question from the one which you asked, so I'll now try the one you've posed.

[Gavin Chipper]

906250?

[Howard Somerset]

My first go gives 959040.

[Andrew Feist]

Working: Suppose the numbers drawn are 123456. Then your phone number must contain at most one of the numbers on the list, so it must be made up of all {17890}, or all {27890}, or all {37890}, or all {47890}, or all {57890}, or all {67890}. Each of these has 5^6 possibilities, so 6*5^6 is 93,750. However, the phone numbers using just {7890} are counted six times, which is five times too many, so we have to subtract 5*4^6 to get a total of 73,270. Subtracting from one million gives 926,730.

Also: I thought the deal was that you couldn't use a repeated number in your phone number twice unless it came up on the show twice. Granted it doesn't matter for this puzzle.

[Gavin Chipper]

I think mine's wrong by the way.

[Jon Corby]

I agree with Feisty's answer. And double-checked using brute force.