Puzzles for Lawyers

One day we may all face the necessity of hiring a lawyer. If the case is tricky the lawyer must be smart and inventive. I am collecting puzzles to give to a potential lawyer during an interview. The following puzzle is one of them. It was given at the second Euler Olympiad in Russia:

At a local Toyota dealership, you are allowed to exchange brand new cars. You can exchange three Camrys for one Prius and one Avalon, and three Priuses for two Camrys and one Avalon. Assuming an unlimited supply of cars at the dealership, can collector Vasya exchange 700 Camrys for 400 Avalons?

The beauty of this puzzle is that the answer I may find acceptable from a mathematician is not the same as I want from my future lawyer.

Have I intrigued you? Get to work and send me your solutions.



  1. Jonathan:

    California leads the nation in lawyers. New Jersey, on the other hand, leads the nation in toxic waste (they had first choice).

    I e-mailed you rather than spoil this. I am confident! But I am not sure, and anxiously await the answer and some discussion of this unusual question.

    I am assuming, by the way, that we are not allowed to borrow someone else’s car for a few minutes. That would change my answer.

  2. Jonathan:

    So, I am “sending” my solution (I mistakenly “sent” it via e-mail.

    By the way, I made up interesting plurals to amuse myself. They are, undoubtedly, wrong:

    By “value” yes, 700 for 400 works, but the exchange is impractical, some annoying parity gets in the way.

    The simplest explanation: Every exchange that gives Avalones also gives an unwanted Camry or Prius. We can’t end an exchange “cleanly” in Avalones.

    Here is one dead end:

    700 Camrys

    Trade 3C for 1P+1A 210 times
    70 Camrys, 210 Avalones, 210 Prii,

    Trade 3P for 2C+1A 70 times
    210 Camrys, 280 Avalones

    Trade 3C for 1P+1A 70 times
    350 Avalones, 70 Prii

    Trade 3P for 2C+1A 21 times
    42 Camrys, 371 Avalones, 7 Prii

    Trade 3C for 1P+1A 14 times
    385 Avalones, 21 Prii

    Trade 3P for 2C+1A 7 times
    14 Camrys, 392 Avalones

    Trade 3C for 1P+1A four times
    2 Camrys, 396 Avalones, 4 Prii

    Trade 3P for 2C+1A once
    4 Camrys, 397 Avalones, 1 Prius

    Trade 3C for 1P+1A once
    1 Camry, 398 Avalones, 2 Prius

    And we are close, and stuck. We didn’t need to review the exchanges to know that this would happen, but it is interesting to complete one version as an exercise.

  3. Jonathan:

    So, from my previous comment, we can reach 1 Camry, 398 Avalones and 2 Prii.

    Borrow 2 Camrys.
    3 C, 398 A, 2 P

    Exchange 3C for 1 A and 1 P
    399 A, 3 P

    Trade 3 P for 2 C and 1 A
    2 C, 400 A

    Return the borrowed Camrys.
    400 Avalones.

  4. Onyebuchi:

    Hm, I’m getting value-wise this should not work. I got the ratios between the number of avalons, A, and the number of camries, C, should be 7/4. That is A/C =7/4. But that appears to have the ratio inverted.

  5. Eamonn:

    Without the need to borrow (as per Jonathan) you can reach the same goal just by simplifying the equations.

    We are given:

    3C = 1P + 1A
    3P = 2C + 1A

    (where C = Camry, P = Prius, A = Avalon)

    Multiplying the first by 3 gives:

    9C = 3P + 3A

    Substituting for 3P

    9C = 2C + 1A + 3A
    7C = 4A

    Which gives you exactly 700 to 400 translation. So the answer would be that yes, you can exchange 700 Camrys for 400 Avalons but you need to exchange them in batches of seven.

  6. Jon (another one!):

    As a side question, can we trade those 700 Camrys in for a whole number of Priuses (560, I believe), instead? By value, yes, but ‘lawyerly’ no, as we have no mechanism for trading in Avalons. Perhaps if we said “You can trade 3 Avalons for one Prius and four Camrys” it would be possible…

  7. Tanya Khovanova:

    Toyota decided that the plural of Prius is Prii:

  8. Jonathan:

    Thanks Tanya! Makes me feel smart. Did they confirm Camrys and Avalones as well?

  9. Vishaal K:

    Just a comment on the reasoning that mathematically it isn’t possible. Let operations X and Y be

    X : 3C -> 1P + 1A
    Y : 3P -> 2C + 1A

    It’s been pointed out that from these relations one sees that 300 X operations and 100 Y operations are required. But I think a crucial detail required for the contradiction is missing: for every Y, there must be 3 X’s preceding or else we’d have a negative number of Prii at some time. Thus the last operation must be a Y and so the previous number of Camry’s is -2 which is a contradiction.

    At $60 dollars an hour, I wouldn’t ask a lawyer. ๐Ÿ™‚

  10. Vishaal K:

    … so what does the lawyer do? ๐Ÿ™‚

  11. mashplum:

    The lawyer offers 900 Camrys for 300 Prii and 300 Avalons. The dealer agrees. The lawyer then insists instead of the 300 Prii he be given 200 Camrys and 100 more Avalons. The dealer agrees. The lawyer then casually suggests the dealer not bother to supply the 200 Camrys but rather to deduct them from the 900 Camrys owed.

  12. Beatriz Y:

    A typical lawyer would first inquire what evidence you have that Vasya has any right to collect the cars. If he/she is satisfied that the right exists, then he/she would say in principle “yes” to the Math question (700 Camry’s may be exchanged for 400 Avalons). In order to understand how he/she reached such conclusion, you would have to pay his/her fees. If the matter becomes a court dispute, then the lawyer would hire a mathmatecian to explain the calculation, which would certainly be “yes” to the question.

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