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!


About Edmark M. Law

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. I blog at You can find me on Twitter @EdmarkLaw and Facebook. My email is
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49 Responses to Ambiguous Profit

  1. Pingback: Learn Fun Facts’ 2017 Year In Review & 1st Year Blogiversary | Learn Fun Facts

  2. Reblogged this on Site Title and commented:


  3. Thanks for the encouragement of liking my first post on here 🙂 It brought me to you. Loved reading your stuff, Good stuff!

    Liked by 2 people

  4. Smitha V says:

    Welcome back. Good to know everything’s settled on the personal front. Will read this one out to my girls’

    Liked by 1 person

  5. Pingback: Ambiguous Profit | The Continuing Adventures of Jimmy Track

  6. selviafei says:

    Hahaahahaha.. the highlight for me is that’s not my bicycle anyway.. 😅😅😅

    Liked by 1 person

  7. 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

  8. cgdtaylor says:

    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.


  9. spleendingo says:

    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.


    • 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.


  10. This is the type of creative thinking that helps brains grow and develop. Thank you for sharing.

    Liked by 2 people

  11. I choose answer ‘D’, which was not provided. He was given the bike in the first place and therefore made $15 profit. 😀 I am too tired for brain benders today. 😉

    Liked by 1 person

  12. natuurfreak says:

    Bedankt om mijn blog te volgen Jouw blog bzv


  13. herbeaton says:

    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

  14. Jenna says:

    I am with Hollywillwander on this. I, too, am a math fence post.
    Though, I must admit, I tried to figure it.

    Liked by 2 people

  15. cindy212 says:

    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

  16. Thanks for following my photo-blog ❤ 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

  17. LightningEllen says:

    Welcome back! Sorry you had to deal with some personal issues but I’m glad all is well now 🙂

    Liked by 2 people

  18. Thank you so much for following me, I’m honored! I’m impressed by your writings and I’m looking forward to read more of your work.
    Best regards,

    Liked by 2 people

  19. Nice to have you back!

    Liked by 2 people

  20. Andrea Frazer says:

    Totally going to read that to my son

    Liked by 2 people

  21. comfykittea says:

    Welcome back! 🙂

    Liked by 2 people

  22. Sandi says:

    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

  23. Good to see you again, 8mark! This puzzle makes me feel two-tired! 「(゚<゚)゙??

    Liked by 2 people

  24. Benjay says:

    Just end the blog since no one noticed you were off? Take back your personal time. Blogging is stupid.

    Liked by 1 person

    • I think that ending my blog just because “no one noticed me” is more stupid. That’s only my opinion of course.

      Liked by 5 people

      • Benjay says:

        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

  25. Garfield Hug says:

    Welcome back!! Glad all is well at your end! Now you have got my brain all boggled😝

    Liked by 2 people

  26. Welcome back! I’ve heard this riddle before – it made my head hurt then too! 😄😉👍

    Liked by 2 people

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