The following is a puzzle I found while reading Perfectly Reasonable Deviations from the Beaten Track by Richard P Feynman.

There are 2 shops which sell oranges.At Shop A you get 2 oranges for 5 cents.At Shop B you have to pay 3 cents for each orange.Richard bought some oranges from both the shops.He totally spent 19 cents.How many oranges did Richard buy?How many did he buy from each shop?

Note that Richard did NOT buy 7 oranges.I tried all the conventional methods I know and still ended up with 7 but it's specifically mentioned that the answer is not 7.

Thanks in advance.

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    $\begingroup$ At the risk of sounding facetious: 6. -2 from Shop A and 8 from Shop B for a total of -5 + 24 = 19 cents. $\endgroup$ – kahen Aug 9 '11 at 4:59
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    $\begingroup$ Are you sure it isn't a typo or some stupid linguistic trick? You are right that the only solution to $5a + 3b = 19$ where $a, b$ are positive integers is $a = 2, b = 3$ (since we have $b \equiv 3 \bmod 5$ and $b = 8$ is too large)., so Richard must have bought $4$ oranges at the first shop and $3$ oranges at the second shop. Any other answer is a cheat. Can you tell us what page the puzzle appears on? $\endgroup$ – Qiaochu Yuan Aug 9 '11 at 5:05
  • $\begingroup$ @Qiaochu: See the link in my answer. $\endgroup$ – joriki Aug 9 '11 at 5:10
  • $\begingroup$ It's even "better" if we allow our shopper to buy negative oranges from Shop B. Then you can use the price difference to buy an unbounded number of oranges. E.g. $-7-5k$ oranges from Shop B (leaving him with $19 + (-3)\cdot(-7) + (-3)\cdot(-5k) = 40 + 15k$ cents) and ${40 + 15k \over 5/2} = 16+6k$ oranges from Shop A for a grand total of $9+k$ oranges. [Insert something about silly questions and silly answers here :-)] $\endgroup$ – kahen Aug 9 '11 at 5:22

I think you misunderstood what Feynman was saying. If you're referring to the February 29, 1944 letter to his mother: He says that unfortunately he didn't have any problems whose answer was $7$ oranges, and then he makes one up whose answer is $7$ oranges; as far as I can tell he doesn't say anywhere that the answer to this one isn't $7$ oranges.

  • $\begingroup$ Yes, I just found that page and I agree with your interpretation. $\endgroup$ – Qiaochu Yuan Aug 9 '11 at 5:15
  • $\begingroup$ @joriki Yes, you do have a point.I think I misunderstood.But even then, is it possible? Have you considered the idea of selling oranges? $\endgroup$ – Green Noob Aug 9 '11 at 5:18
  • $\begingroup$ @Green: not unless you do something outside of the implicit confines of the puzzle, and there are a million ways to go outside the confines of the puzzle. $\endgroup$ – Qiaochu Yuan Aug 9 '11 at 5:21
  • $\begingroup$ @Yuan Just for curiosity's sake can you mention one of them? $\endgroup$ – Green Noob Aug 9 '11 at 5:26
  • $\begingroup$ @Green: kahen considered the idea of selling oranges in a comment under the question. Other ideas mentioned in a now-deleted answer were that one of the shop owners may have given him a discount, or he may have spent money on other things. $\endgroup$ – joriki Aug 9 '11 at 5:41

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