# Videogame salesmen

Two gamers sell their collection of XBOX games. They've got as many Euros as there were games in the collection originally for each game.

They split the money by having the first gamer take 10 Euros from the pile of money, then the second takes 10, etc. until the pile is gone, but the first gamer came to a take where there was less than 10 Euros left.

So, the first gamer took what remained and gave the second gamer his old XBOX controller to make the split fair.

How much was the controller worth?

The price per game is the same as the number of games, so the amount of money they got is a square number. When splitting the money they both take out 10 the same number of times because the first taker is the one who hits the small remainder. Therefore they both have taken $10k$ for some $k$ and the first person takes the remainder $r$, which means the total amount of money is a square number of the form $20k+r$ with $0<r<10$. The first person has $r$ more than the other, so to even the money out, he has to give away something of value $r/2$. Assuming the XBOX controller is worth a whole number of Euros and is wholly owned by the first person rather than being shared property, then it must be the case that $r$ is even. The even square numbers modulo 20 are 0, 4, 16. Therefore $r=4$, and the controller is worth half that, i.e. $2$ Euro.

Edit:

The solution hinges on the fact that $r$ is even. I derived that fact by assuming that the XBOX controller is worth a whole number of Euro's. It is also possible by using a different assumption, namely that the collection of games contains an even number of games, because both players contributed the same number - after all, they split the money evenly. My impression was that the games collection was under shared ownership so that it could contain an odd number of games, and hence the assumption on the controller's price was necessary.

• Yes, 2 Euro. Good job! That's also the first gamer's moral debt. – user39732 Aug 26 '17 at 22:55

Whilst Jaap Scherphuis' solution certainly works, the question doesn't specify that the controller is worth a whole number of Euros and so there are (at least) three other possibilities:

They sold 5 games for 5 Euros each, for a total of 25 Euros. After one turn of each of them taking 10 Euros, they have each taken 10 Euros and there are 5 Euros left. If the first gamer giving the second gamer his controller makes things fair, then the controller is worth half of 5 Euros, which is 2 Euros and 50 Euro cents.

or

They sold 7 games for 7 Euros each, for a total of 49 Euros. After two turns of each of them taking 10 Euros, they have each taken 20 Euros and there are 9 Euros left. If the first gamer giving the second gamer his controller makes things fair, then the controller is worth half of 9 Euros, which is 4 Euros and 50 Euro cents.

or

They sold 11 games for 11 Euros each, for a total of 121 Euros. After 12 turns of each of them taking 10 Euros, they have each taken 60 Euros and there is 1 Euro left. If the first gamer giving the second gamer his controller makes things fair, then the controller is worth half of 1 Euro, which is 50 Euro cents.

Edit: Wen1now, clarified by Jaap Scherphuis, suggests that the question implies (without explicitly stating so) that each gamer contributed an equal number of games to the collection, which would mean the collection would have to contain an even number of games, and thus that none of these solutions would work. That's a good point, but on reflection I think, as Jaap Scherphuis and Greg Petersen both opine (if I understand them correctly), that this need not be the case. My solutions rely on the gamers having an equal stake in the collection (an assumption I hadn't realised I was making till Wen1now's comment, since I hadn't noticed it wasn't given as part of the question, so thanks for that), but it could be for example that the collection has always been communally owned, with each of the gamers chipping in an equal amount to purchase each game (or regularly chipping in equal amounts to a game-purchasing fund) and then sharing ownership. In that case an odd number of games is certainly possible. (That might not sound so plausible considered as a real-life situation rather than a puzzle set-up, but I can imagine it happening with two gaming siblings.)

• I believe there's the assumption that both gamers donated the same number of games to the selloff. – Wen1now Aug 28 '17 at 6:59
• @Wen1now You're absolutely right! I was assuming that, and hadn't noticed it wasn't stated in the question. Thanks for pointing that out - have now edited to include it. – Neremanth Aug 28 '17 at 15:09
• I think @Wen1now's point is that if they both contributed the same number of games, then the total number of games will be even. I disagree though. I assumed that they both had shared ownership of the set of games, which could then in principle contain an odd number. – Jaap Scherphuis Aug 28 '17 at 15:27
• The question is worded as "Their Collection", while a subtle point, I think this means it is a joint collection. Otherwise I think it would have been worded as "Their Collections" – Greg Petersen Aug 28 '17 at 16:04
• Ah, thanks @JaapScherphuis, that makes sense! With you (and @Wen1now) now, I think. It's a good point but I agree with you and @GregPetersen; have edited to discuss. – Neremanth Aug 28 '17 at 21:42