Ambiguous Profit


It’s been quite a while isn’t  it? In my last post, I said that I’d be gone for a week but it turned out to be almost four months. During the last few months, I had to deal with some personal problems. I’m just glad that it’s all over now. So, I apologize for not informing you about my unexpected long absence, though during those times, blogging and the Internet as a whole were the last things on my mind.

But the most important thing now is that I’m back and I’m now excited to blog once again.

I found a curious puzzle in Sam Loyd’s Cyclopedia of 5000 Puzzles, Tricks and Conundrums, with Answers (1914).  While several of the riddles and short puzzles from the book are either dated or mediocre, many of the puzzles in it are still great even by today’s standards.

The following puzzle is one of the few puzzles in the book in which Loyd didn’t provide a straightforward solution.

A dealer sold a bicycle for $50, and then bought it back for $40, thereby clearly making $10, as he had the same bicycle back and $10 besides. Now having bought it for $40, he resold it for $45, and made $5 more, or $15 in all.

“But,” says a bookkeeper, “the man starts off with a wheel worth $50, and at the end of the second sale has just $55! How then could he make more than $5? You see the selling of the wheel at $50 is a mere exchange, which shows neither profit nor loss, but when he buys at $50 and sells at $45, he makes $5, and that is all there is to it.”

“I claim,” says an accountant, “that when he sells at $50 and buys back at $40, he has clearly and positively made $10, because he has the same wheel and $10, but when he now sells at $45 he makes that mere exchange referred to, which shows neither profit nor loss, and does not affect his first profit, and has made exactly $10.”

It is a simple transaction, which any scholar in the primary class should be able to figure out mentally, and yet we are confronted by three different answers! Which in your opinion is right?

While this question may spark some interesting debates and arguments, the truth is this question can’t be answered without a vital information – the price the dealer paid for the bicycle when he first bought it. In other words, you won’t be able to calculate the profit without first knowing the real monetary value of the bike in the first place.

If you want to be philosophical about it though, you may either define the value of the bicycle based on the first, second or third transaction. If you choose the first transaction ($50), then the “profit” would be $5. For the second transaction ($40), the “profit” would be $15. Lastly, if you choose the third transaction ($45), the “profit” would be $10.

Of course, those are just all assumptions since there is no way for us to figure out how much it cost the dealer when he first obtained the bike. For all we know, the bike may have been given to the dealer as a gift!

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My name Edmark M. Law. I work as a freelance writer, mainly writing about science and mathematics. I am an ardent hobbyist. I like to read, solve puzzles, play chess, make origami and play basketball. In addition, I dabble in magic, particularly card magic and other sleight-of-hand type magic. I live in Hong Kong. You can find me on Twitter` and Facebook. My email is

50 thoughts on “Ambiguous Profit

  1. Cool story! In my opinion, there are two vital pieces of information. 1) We don’t know what the bike is actually worth (as you point out). Suppose the bike is *actually* worth $200! If he ends up without the bike, he has lost value (though not necessarily cash). 2) We don’t know the value of his use of the bike. Suppose he had a bike and used it for five years every day and saved $2000 worth of commuting expenses? Wouldn’t he have gotten a lot more value out of that bike than if he had kept in the garage for five years? I think there is a deeper problem revealed by this story: we tend to think in terms of dollars and not in terms of value. Money is a great medium of exchange, but it isn’t always infinitely fungible. When my wife and I moved to California, most of our “stuff” was burned up in a moving van fire. Among the many things lost were souvenir refrigerator magnets that we collected on our travels. It would be virtually impossible to replace them, but if we could, it would require thousands of dollars of travel. Even so, it wouldn’t be the “same” because the originals were purchased as part of a trip that had meaning. That meaning would be much different and lesser if we simply travelled in order to purchase refrigerator magnets!

    Liked by 1 person

  2. Thanks for this puzzle.

    I remember something similar being told at the time of the banking crisis. Basically, a $10 note gets spent and re-spent with the result that multiples of $10 go into the economy.


  3. I believe the price of anything, despite current market values, is based solely on one thing.

    The only true “worth” of anything is: how much is someone willing to pay for it? If I see something that is of value (for sentimental or other reasons), I might pay more than another person.

    Deciding value based on a “list” or “guide” is only an opinion of an objects true value, which is how much is it worth to the buyer.


    1. Indeed, that’s why people can sell seemingly worthless things at high prices.

      However, that’s not the point of this puzzle. The point is, in order to determine the profit of the dealer, the information about the price of acquisition of the bike (the amount of money the dealer paid) should be known. However, this information was (intentionally) ommitted in the question. Therefore, the question can’t be objectively answered.


    1. If that’s the case, then he would have made $55 of profit since acquiring the bike didn’t cost him anything.

      50 – 40 + 45 = 55

      By the way, I also mentioned in the last paragraph the possibility that the bike was given to him as a gift

      Liked by 2 people

  4. one must remember the famous 3 interviews for a tax accountant job. “What is 2 + 2 ?” Answer 1st interviewee “4”…no job, 2nd interviewee “4”…no job, 3rd interviewee “what would you like it to be?” gets the job.

    Liked by 2 people

      1. No, I just gave myself a headache and used three sheets of paper. Stubbornness didn’t help, but my friend saw what I was doing and said, “What was the original cost of the bike?”
        I said, “Fifty dollars.”
        “They said, “No. What did the merchant buy it for to start with?”
        I said, “Uh… I dunno.”
        I think he was trying to help, but I threw the papers at him and stormed to my room.

        Liked by 2 people

  5. Hi
    I read paragraph 1 over and over until I realized I was merely halfway through a cup of coffee. Heck, I considered getting a ledger out. Then I thought, what’s the initial balance (what the bike cost). But phttp. I am not a math gal)?

    Liked by 1 person

  6. Thanks for following my photo-blog <3 I really appreciate it! For someone who is about as math-brained as a fence post (that would be me), I gotta admit this was good although I gotta admit my head was tilted a little after reading it and my brain isn't sitting in the same place either…

    Liked by 1 person

  7. My husband is a math guy (aerospace engineering degree) and loves problems. I hate these things. But, it’s funny, this can be applied to the basic of basics. We see this happening when we have those online yard sales. Here’s a group of books just $10.00. All the kids books originally cost $6.00 (they show you the back cover with price) So you’re getting a “steal.” But some of those books were given to them or purchased at the dollar store. Sometimes they are trying to make a profit. Not the purpose of those Facebook group sites. They clearly state no buying something on sale, and trying to sell at higher amount. That is not what those sites are about…yet, you know people are doing it.

    Liked by 2 people

      1. It is your reaction more than formed opinion. It was funny. Thanks. Mine is the popular response of the unblogging slightly cynical masses? You could offer an opinion on my reaction? My reaction is infotainment? Your blog is well done. If not, I would not read it. Thanks. Great response.

        Liked by 2 people

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