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Unfair Mensa test -- Horsedealer Problem

You should have gotten to this page from here.
If you didn't, this won't make quite as much sense.

First, let me reproduce that section.

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§     Problem #2 in Test 6 admits of more than one perfectly correct answer, which should not be the case.


2.  A man bought a horse for £20 and gave in payment a cheque for £30.  The horsedealer persuaded a shopkeeper to change the cheque for him, and the buyer, having received his £10 change, rode off on the horse and was not seen again.  Later the cheque was found to be valueless, and the horsedealer had to refund the shopkeeper the amount he had received.  The horsedealer had himself bought the horse for £10.  How much did the horsedealer lose altogether?

(1) Nothing.     (2) £20.      (3) £10.     (4) £30.      (5) £40.

It is not made clear at all what "How much did the horsedealer lose altogether?" specifically means.  It could be taken as meaning,

"How much did the horsedealer lose altogether in cash?"

or it could mean

"How much did the horsedealer lose altogether in cash and lost profits?"

or it could mean

"How much did the horsedealer lose altogether in cash and lost profits and horses?"

If the question is how much altogether in cash, then the answer is that he lost £40, consisting of the £10 in change that he paid to the thief plus the £30 he refunded to the shopkeeper.

If the question is how much he lost altogether in cash and lost profits, then the answer is is the same £40 from the interpretation above plus another £10 for the profit he lost on the sale, for a total of £50, since he bought the horse for £10 and would have sold it for £20.  Notice that this legitimate answer, of £50, is not even offered as one of the possible answers.

If the question is how much he lost altogether in cash and lost profits and horses, then the answer is the same £50 from the interpretation above plus another £10 for the loss of the horse itself, for a total of £60, since the horsedealer now has an inventory deficiency of paid-for horses to the tune of one £10 horse.  Notice that this legitimate answer, of £60, is also not even offered as one of the possible answers.

I answered £40 because that is the largest amount you offer as a possible answer, but I object to the wording of the question.  And if the correct answer is indeed £40, how do you or the test writer respond to my arguments for £50 and £60 as answers?

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Now, as I said on the previous page, I did make a mistake.  (I maintain that the test question admits of more than one legitimate answer, but not more than two.)  My mistake was pointed out to me, in varying degrees of accuracy and clarity, by several e-mail correspondents over the last year or so.  The best of those e-mail conversations is shown below.


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Subj: Horsedealer problem
Date: 2/5/2001 1:28:01 AM Central Standard Time
From: Daniel J. Hulme

I am not currently a Mensa member, but I have always scored between 150 and 170 on all IQ tests that I have taken.  The very fact that I have to state a range for my IQ scores already points to one inconsistency in the tests: Two "impersonal, scientific" tests cannot give the same results.  The magnitude of error for me has been between 12% and 13%.  With that out of the way, I would like to address your concerns.

One of your concerns was that the writers of the test obviously were lacking in their instructions: You say, "The instructions you provided are both imprecise and, in some cases, contradictory."

You show that the explanation for the "Classification" part contains a semantic error (referring to mammals simply as animals).  I prefer to believe that it was simply a wording mistake, but the mistake was in the explanation, not in the question itself.  It is true that these things should be very specific, but I don't think this small wording error has caused anyone real harm.  The definition of mammal is as follows


Definition (2) in both dictionaries implies what many people do not know, which is that, even though it's scientifically inaccurate (kingdom animalia contains nearly all forms of life, including Aves, or birds), the word animal is often used synonymously with the word mammal.

Thus, we can see that either the writer made a wording mistake or he was simply using the word animal to mean mammal.  However, I would agree with you that these things ought to be less confusing and perfectly accurate. It would be an improvement to use the word mammal.  However, it is practically irrelevant to the test, since it was not part of a question.

The instructions you cite thereafter are indeed contradictory, although I think that the exceptions to the rule state themselves quite clearly as such: "Indicate more than one if more than one satisfy the required conditions," "Place a cross against the two sums of money that he entered up wrongly."

The instructions for the test would better have read, "Each question has only one correct answer unless otherwise specified."

The question relating to the well-known riddle should definitely not have been included, as it gives an unfair advantage for simply having heard it before.  Ideally, IQ tests would not use already-known problems.

So, thus far, I would have to agree to your conclusions, although I do not think they are as drastic as you seem to think.  This test obviously was not as well thought-out as it could have been.

However, with regards to the problem of the horse sale, I think there is only one possible answer that makes sense.  This is my logic on the problem: We will not consider the horse to be anything but property (i.e., it is not equal to its possible selling value, as this could change).  However, it should be included in the losses/gains.

Let us assume that the horsedealer starts with an arbitrary amount of money, say £10.  The difference at the end will show exactly how much he lost.


Starting point >>   £10
Pays for horse - £10 £ 0
Receives horse + horse horse + £ 0
Receives check + check horse + check + £ 0
Cashes check - check horse + £ 0
Gets cash for check + £30 horse + £30
Gives £10 in change - £10 horse + £20
Gives horse - horse £20
Refund for bad check - £30 - £10
Ending point >>   - £10

Since the man originally had £10 and ended up with -£10, he lost only £20.  Yes, he did lose his horse, but the value of the horse is in question (it is only worth what he can sell it for).  Therefore, he gained a horse, and he lost a horse.  You could argue about the possible gains (i.e., he could have sold the horse for £20, and therefore it was worth more than the £10 he paid for it), and say that he lost £30 when losing the horse worth £20.  However, the £30 is irrelevant, since he essentially borrowed it and gave it back.  All he lost was the horse and the £10 he gave in change.  This would make the answer either £20 or £30.  I believe that only £20 makes sense since, from beginning to end, he came out £20 pounds behind.

Dan, I don't understand why the fact that the value of the horse is in question means it has no value.  The value of the net assets of St. Peter's Basilica in Vatican City is in question, but surely it's not zero.

In the case of valuing the horse, it would be contrary to generally accepted accounting principles to assume the value of an asset is nil.  Unless there's some reason not to, one would always account for the value of an item of merchandise at cost, in this case £10.  You seem to have assumed that because it's not exactly known, that value is zero.

This includes your two steps of including lost horses and lost profits.  If we include the lost profits, it is £30, and otherwise it is £20.  As I said, £20 makes more sense since he had a potential of making a profit on the horse, but no guarantee.  Therefore, at the beginning he lost £10 and gained a horse.  He was left with a horse, the worth of which is in question.  When he later lost that horse, he didn't lose £20 or £10; he just lost a horse, leaving him with a deficit of £10, since he earlier paid that sum.  The only other money he really lost was the £10 in change he gave to the buyer, which gives a deficit of £20.  This is the correct answer.

Your fault in logic is considering the £30 as lost money, when in fact he first received this money, then returned it.

All of the tests I have taken have had discrepancies, mis-wordings and confusing questions. Some have been better than others.  However, I wanted to point out the flaws in your own logic as well as tend to agree that these tests are not perfect.


Dan Hulme


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Update of February 9, 2011: Just so you'll know, I have received many, many emails in response to the horesdealer problem over the last decade, and I do read and respond to all of them, and I will continue to do so, but it wasn't till today, ten years later, that I received another one worthy of reproduction on this page.  Here it is in full.

Subj: Horsedealer problem
Date: 2/9/2011 2:31:46 PM Central Standard Time
From: Peter Klimczak

There is a much easier way to analyze the horesdealer problem.  The shopkeeper lost nothing.  The thief paid nothing and gained the horse and £10.  The horse cost the horsedealer £10.  So the horsedealer lost £20.

I'm pretty sure this is right, and I'm quite sure it's easier.  It starts with a critical but unstated assumption, which is that whatever the thief stole is what the shopkeeper and the horsedealer lost.  Since the shopkeeper broke even, that leaves only the horsedealer to suffer any losses caused by the thief.  The thief gained a horse worth £10 plus £10 in cash, which means the horsedealer's losses totalled £20.  Same answer, with a little less math.

I replied to Mr. Klimczak as follows:

"I would still like to argue that the horsedealer lost not only a £10 horse and £10 in cash but also £10 in profits, for a total of £30.  Said another way, had the thief's cheque not been valueless, the horsedealer not only wouldn't have lost £20, he would have gained a profit of £10.  The question was how much he lost altogether."

His reply:

"Itís a valid argument.  The fact there even is an argument makes it a bad (or badly phrased) question for Mensa, which was your original point.  It seems how much the horsedealer lost is more a matter of semantics and psychology than logic.

"What would you answer to the following question: 'A horsedealer bought a horse for £10.  He put it up for sale at £20.  The next night it was stolen. How much did the horsedealer lose?'"

My reply:

"Does your analysis account for his lost profit?  As I see it, how much he lost altogether turns on the only extraordinary event, which is that the cheque was valueless.  Had it not been, the horsedealer would have gained a $10 gross profit on the sale of the horse."

His reply:

"I assume the buyer was crooked and knew the cheque was worthless, so it was not an 'extraordinary event' but integral to the scenario.  He was basically a thief.  But itís arguable."

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Update of June 18, 2011: An email correspondent has a really new take on the meaning of the question "How much did the horsedealer lose altogether?"  He takes "lose altogether" to mean all the assets the horsedealer ever owned in this situation that were ever transferred to another party.  The term he uses is "outgoings."  By this interpretation the horsedealer lost the following:

-- the £10 he paid for the horse,
-- the £10 in change he gave the thief,
-- the £30 he refunded to the shopkeeper,
-- the £10 horse, and
-- the £30 cheque,

for a grand total of £90.

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Update of March 2, 2012: Below is the second-longest email I have ever received on the horsedealer problem.  It just goes on and on, but it's worth it or I wouldn't have taken the time to reproduce it here for you.  You'll find Tim Hernly's analysis to be both thorough and lucid.  Happily, it's also well-written; the only editing I did was utterly insignificant, such as changing "30" to "£30."

You may disagree, but Mr. Hernly concludes that the answer to the question "How much did the horsedealer lose altogether?" is £30, not the £20 asserted by Dan Hulme and later by Peter Klimczak above.  The fact that Mr. Hernly's conclusion agrees with mine is, I swear, not one of the reasons I chose to offer his analysis for your consideration.

Subj: Horsedealer problem
Date: March 02, 2012 6:12 PM
From: Tim Hernly

Hey, Johnny,

I would like to assist you with this question.  Let me first clear the air with a disclaimer.  I am not a member of MENSA.  Also, I don't know my IQ and if I did I wouldn't air it in public.

Joining an organization such as MENSA strikes me as not dissimilar to feeling pride for being a descendant of someone who sailed aboard the Mayflower or feeling shame for having been born with a low IQ or a birth defect.  This is not a slam against anyone who is a member; I'm sure there are many smart, interesting, kind and caring, if misguided, people who belong to MENSA.

Anyway, based on the facts posited in the horsedealer question, your (revised) answer of £30 is the only correct answer among the choices offered.  There a very easy way to get to the £30 conclusion, which I'll address later.

First, some analysis of the question "How much did the horsedealer lose altogether?"

The question is indeed ambiguously worded but presumably it refers to the horsedealer's ECONOMIC loss on THIS TRANSACTION.  He lost cash and a horse and perhaps his temper, pride, or sanity, but this by its context is a question of logic and economics.  Importantly, it's an economic question and NOT an accounting question.  These are two different things.  Answering an accounting question requires reference to some specified accounting conventions, which vary dramatically in purpose and operation across cultures, time and individual business enterprises.  We are not told which set of accounting standards or principles or methods to apply, and rightly so, because this question was not designed to test one's skill at applying accounting conventions.

This is also NOT a question about the loss incurred by the horsedealer since the beginning of time; it refers only to this particular sale transaction.

OK, so the stage is set.  It's clear the horsedealer lost £10 net cash on the sale of the horse (+30 - 10 - 30 = -10) AND he lost the horse itself, as measured by its market value.  So the horsedealer's total loss equals the sum of the lost £10 cash plus the market value of the horse.

The facts do not explicitly state the market value of the horse per se, but a market value of £20 is implied by the fact this was the price negotiated between unrelated parties, neither of whom was under a compulsion to buy or sell, each acting in his own self-interest.  This defines the very process by which market value is derived.

One might object that using a £20 market value in this case is dubious, that the buyer was clearly not a true arm's-length participant operating in an open market, because he was obviously a fraudster intent on ripping off the horsedealer, and thus the agreed £20 price was probably inflated.  But this objection suffers three problems.

First, the facts do not state the buyer knew at the time he agreed on the price that the £30 cheque was not backed by sufficient funds.  The facts do not even state that the cheque was written on the buyer's own account.  All we know is that the buyer had a £30 cheque in his possession, which he endorsed over to the horsedealer and the horsedealer in turn endorsed over to the shopkeeper and the shopkeeper in turn endorsed over to the bank and the bank rejected.  The fact the horsedealer never saw the buyer again might be explained any number of ways.  We cannot add facts not posited by the problem.

Second, even assuming arguendo that the buyer knew the cheque would bounce, that buyer was indeed motivated to negotiate the lowest price he could for the horse, just as any other open-market buyer would have been.  If he had struck the bargain at £18 rather than £20, he would have received £12 in change for his £30 cheque rather than only £10.  And of course the horsedealer was motivated to sell at the highest price.  Market forces were in place to hammer out a market-value price of £20 for the horse.

Third, there is not a shred of other evidence of the horse's market value stated in the problem that could lead one reasonably to conclude an alternative answer.  Whether the horsedealer paid zero or £10 or £1,000 for the horse makes utterly no difference.  The problem does not say whether the horsedealer had bought the horse as a foal three years earlier, or that he bought it the same day.  It does not tell us how much in expenses he had paid feeding and otherwise tending the horse prior to its sale.  It doesn't tell us the circumstances of the horse's acquisition by the horsedealer.  The horsedealer's original, historical cost of the horse of £10 is a red herring.  To illustrate this point, assume the horsedealer had paid £1,000 for the horse a year earlier and sold it for £20.  That £980 loss in value would have been incurred independent of this sale transaction, due to some earlier change in the horse or in market forces.  His loss ON THIS TRANSACTION would remain the same.  Again, don't confuse the accounting for the transaction with the economics of the deal.  When the horsedealer should "recognize" that loss by recording the "cost of goods sold" amount on his books is an accounting question and depends upon the conventions or rules to be applied.

All that matters to answer this question is the market value of the horse at the time of the sale.  Anyone who is confused by this need only substitute the word "gold" for "horse" in the question and this will become apparent.  The argument that one may ignore the value of the horse in answering this question could not have been propounded by anyone who has operated a for-profit enterprise.

Now for a MUCH easier way to solve the whole problem: The horsedealer is left holding a bad cheque for £30.  That is the value of the horsedealer's claim against the buyer and that is the amount of his loss if, as the posited facts imply, the horsedealer is able to collect nothing on his claim.  It's often easy to overthink these things when a little big-picture perspective, also known as "common sense," will provide the answer.

If this still isn't entirely clear, try the "gold" substitution trick: A speculator buys an ounce of gold for £1,000.  He later sells it for £1,500.  The £1,500 cheque bounces.  How much has the speculator lost?

A truly silly answer would be that he hasn't lost anything because it was just gold, not money.  An equally specious "accounting" answer would be that he has lost only his £1,000 historical cost of the gold.  Common sense economics says he's out £1,500.

Hope this helps.

-- Tim

In a follow-up email the next day, in reference to the analyses preceding his, Tim offered this: "I found it interesting that intelligent persons arrived at such diverse interpretations and conclusions.  The ambiguous phrasing of the facts can account for only part of this result."

I find it interesting that the horsedealer problem has generated so much thought and so much email over the years.  What surprises me is that I don't really care what Mensa's answer is, nor do I care what their reasoning is.

If you disagree with any of the answers above or have a new way of arriving at whatever your answer is, as always, .







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