Five Down

One of my favourite blogs is Futility Closet, which is sadly appropriate given its tagline “An idler’s miscellany”. This week it featured a puzzle called Five Down devised by the English mathematician Henry Dudeney. Since the subject of the puzzle is money, it seems like an appropriate one to share here on the Mule.

A banker in a country town was walking down the street when he saw a five-dollar bill on the curb. He picked it up, noted the number, and went to his home for luncheon. His wife said that the butcher had sent in his bill for five dollars, and, as the only money he had was the bill he had found, he gave it to her, and she paid the butcher. The butcher paid it to a farmer in buying a calf, the farmer paid it to a merchant who in turn paid it to a laundry woman, and she, remembering that she owed the bank five dollars, went there and paid the debt.

The banker recognized the bill as the one he had found, and by that time it had paid twenty-five dollars worth of debts. On careful examination he discovered that the bill was counterfeit. What was lost in the whole transaction, and by whom?

I will not reveal the solutiuon here to give you a chance to think about the puzzle. What I will reveal is that the “solution”, originally published in The Strand in 1917, was re-published on the blog yesterday but it is in fact incorrect! Understanding what is wrong with the original solution (and the blog’s author was quick to provide an update following feedback from his readers) gives some insight into two of the roles money plays: a medium of exchange and a store of value.

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16 thoughts on “Five Down

  1. JamesGlover

    $0. Suppose the counterfeit bill had been swapped for a real bill with no one the wiser at the first stage when the banker picks it up. Then when he receives the real bill back he swaps it back for the counterfeit bill. (Even if he didn’t have any money post-GFC he could have borrowed a real note and paid it back). The transactions are all still the same to the participants and since at the beginning he had nothing and at the end he had a worthless piece of paper so net-net $0. He isn’t really up ’cause he would have paid the bill with a real one he kept in his pocket all along or borrowed the money and paid it back.

    If you want to be finicky: at worst he is up the interest he receives on his real bank account he doesn’t draw on (alternatively up on the interest he doesn’t have to pay to borrow a real bill).

    Of course being an immoral banker he would slip the counterfeit note into a pile of notes the next time he paid for a taxi.

    Disclaimer: I haven’t read the solution so fully expect to be wrong.

  2. Stubborn Mule Post author

    James: I like the swapping analogy. The one I thought of was to imagine the note was in fact a hand-scribbled IOU from the banker. Assuming everyone knew and trusted him (a banker??) all the transactions could take place as with a real $5 and when the banker gets it back he can safely tear it up.

  3. OldFuzz

    I like to work backwards:
    Bank loses $5 accepting the dud for the Laundrywoman’s debt
    The Laundrywoman is square – owed 5 and is owed 5 – both settled
    Ditto the merchant, farmer and butcher
    The wife “owes the banker husband 5 which he “loaned her for the meat
    The banker got the meat which in fact he consumed, making him a Fat Banker
    The net result Banker’s Bank paid $5 for the meat! Fair enough…..

  4. Stubborn Mule Post author

    As has been pointed out to me in off-blog correspondence, if the bank is not actually owned by the banker (i.e. he is just an employee), the banker is up $5 and the bank is down $5 unless the banker chooses to make the bank whole again.

  5. dd

    Looks like all commenters are correct, based on the bank/banker relationship assumptions. Smart correspondents, Mule.

  6. JamesGlover

    @mule – probably when this was originally proposed in 1917 a Banker (in the Monopoly game sense) would definitely have meant a guy who owned a bank. They were risky, unregulated entities run by shifty, greedy individuals then…oh.

  7. Pingback: More on “Five Down” | Stubborn Mule

  8. Ramanan

    Our desire to hold money as a store of wealth is a barometer of our distrust of our own calculations and conventions concerning the future.… The possession of money lulls our disquietude.

    – John Maynard Keynes, 1937

  9. Ramanan

    Yes Sean … I found some good ones

    The difficulty lies, not in the new ideas, but in escaping the old ones, which ramify, for those brought up as most of us have been, into every corner of our minds.

    A study of the history of opinion is a necessary preliminary to the emancipation of the mind.

    As they say – we are all Keynesians now!

  10. Marco aka Cracticus

    Hi Stubborn,

    Finally catching up with interesting stuff!

    For what’s worth, I arrived at Jimbo’s solution considering assets and liabilities: essentially, everybody (except the banker and the bank) had an account receivable of $5 AND an account payable of $5: they both cancel out. If we assume the banker owns the bank, Jimbo’s solution follows.

    But that’s not why I write this. Do you remember a conversation we had some three weeks ago?

    magpie: @sean and MMT gurus: Second question: according to MMT, banks do create money through loans. So from this could we conclude that both, the Government (including central bank) and banks create money. Is that right?

    sean: @magpie yes, but there is an important difference. Banks cannot control that money being destroyed again when a depositor wants to withdraw their money.

    magpie: @sean I don’t understand this. Say, Bank A lends Joe $1000. With that money, Joe pays Fred. The money is still there. It only ceases to exist when Joe pays Bank A the $1000. Isn’t it right?

    sean: @magpie let’s say Joe and Fred both bank with Bank A. Bank A lends Joe $1000 who credits Fred’s account and Bank A has a new asset (the loan) and liability (the deposit which is the “money”). Fred then withdraws as cash, which the Bank has to provide and the bank money disappears

    magpie: @sean Sorry, @sean (this resurrection thing means I’m basically a zombie: ie brainless :o)). I still don’t get it: Joe still owes the money. Besides, what if Joe and Fred don’t bank with Bank A? Anyway, don’t worry.


    Say in the Five Down example: the bank lent the laundry lady $5 (that’s why she owes the bank $5: liability, account payable) and she worked for the merchant for wages of $5, but has not been paid yet (asset, account receivable).

    She also has $5 in her bank account (the $5 the bank lent her) and she presumably does something with that money (say, buys herself some soap).

    Similarly, the merchant (who owes the laundry lady $5: account payable, liability) extended the farmer a credit for $5 (account receivable, asset) and the whole things goes on until we reach the banker.

    Somehow the banker gets some cash (liquid assets, the $5 note he didn’t know was fake) and pays his $5 butchery debt (liability).

    The money only disappears when the laundry lady pays the bank the exact amount she owes the bank. In the example, the physical note itself actually disappears (it’s a counterfeit, so the bank destroys it, or hands it over to the police, or whatever).

    Imagine she had paid less than the $5 she owed. Say, she paid $3. This means that either herself, or someone else, has kept $2 out of the $5 they received. But she still owes the bank $2. If she was paid only $3 by the merchant, then the merchant still owes her $2.

    The money wasn’t created by the laundry lady depositing money in the bank. It was created by the bank lending money to the laundry lady. And, as we know, the bank doesn’t need to have deposits or reserves before being able to lend money.

    And the money would not have disappeared if the merchant, instead of paying $5 to the laundry lady, had simply deposited the money in the bank.

    I know it’s a bit confusing, but if I am mistaken, I can’t see my mistake. Any ideas?

  11. Stubborn Mule Post author

    Marco: re-reading that Mule Stable exchange, I understand your confusion. My replies were confusing at best, conflating a couple of different issues.

    With a bit more space at my disposal, here are a few more thoughts:

    * Financial transactions affect liabilities/assets (of the transacting parties) in an equal and opposite manner. For example, I buy something from you, my assets decrease (or perhaps my liabilities increase if I borrowed money to pay you) and your assets increase (or your liabilities decrease if you apply the money I pay you directly to an outstanding loan). Importantly though, offsetting this is a transfer of real goods/services of the same value (at least as far as the transacting parties assess the value).

    * Many (most?) financial transactions do not involve a transfer of “real” goods and services, but rather modify the nature of the assets or liabilities. For example, if I move money from a transaction account to a term deposit, my net assets/liabilities do not change.

    * Bank deposits are liabilities of the bank and their loans to customers are assets of the bank. When a bank lends money, it is a financial transaction that does not change the net asset/liability position of either party (for now…in the future the borrower may, of course, default). The classic example is that the bank lends me $100 and makes the money available in a transaction account. As a result, the bank’s assets are up $100 (the loan) and liabilities are also up $100 (the transaction account balance).

    * What does all this mean for “creating money”? The tricky thing is pinning down what “money” actually is. Traditionally in the textbooks, money is taken to be notes, coins and transaction account (or “checking” account) balances. In the $100 example above, the $100 in the transaction account is “new” money. The key is that while there is no net change in assets/liabilities, there is a change in money as the liability is considered money and the asset (the loan) is not considered “negative money”. It is in this sense that people speak of loans “creating” money.

    * How can money be destroyed? Basically, through transactions that decrease the size of liabilities that are considered money. For example, if I took the $100 and paid it to you and you used the money to repay your loan with the same bank.

    * The point I was trying to make about transfers to another bank gets at a somewhat different issue. Once people latch onto the idea that loans create deposits, they may wonder why a bank needs to borrow money in the wholesale markets if they can simply create money by lending. The answer is that it if I pay you the $100 and you bank with another bank, that bank will not accept payment from my bank in the form of a balance with my bank. Instead it would require a transfer between the accounts of the two banks with the central bank. If my bank does not have sufficient funds with the central bank, it must borrow them either from the central bank or from another bank either directly or via another investor.

    * Transferring money to another bank may or may not “destroy” money in the system (although it may if it is used to discharge debts), but it does mean that banks are not really in control of money even though they can, in a particular way, create it.

    Hope that makes some sense!

  12. Marco aka Cracticus

    I think it does.

    In fact, either we are arguing past each other (explaining the same thing without realizing it), or there is something too subtle for me to understand, in which case I apologize.

    Let’s go back to the Five Down example. When the bank lends the laundry lady 5 bucks, it deposits the money in the laundry lady’s “checking account” (and this is a liability for the bank as you correctly say). As I don’t dispute that, I believe, up to this point, we are still in agreement.

    I did not mention this in my post (I supposed it was obvious), but the bank’s accountant (who happens to be the bank’s teller, too), 10 seconds after the laundry lady left the bank’s premises, took the old book from under the counter and wrote down two entries in the bank’s book (i.e. balance sheet):

    (1) He created the “Laundry Lady’s checking account” entry in the book (in the liability side, as explained before).
    (2) He created also a “Laundry Lady’s loan” entry in the asset side of the book.

    Thus, with these two entries, the balance sheet balances. Notice, however, that although the bank’s book balances, the bank’s assets (in the form of loans) and liabilities (in the form of the checking account) do increase.

    From the bank’s point of view, it is a liability, because the laundry lady can claim that money for her own purposes. It is an asset, because the bank will claim that money back from the laundry lady at a given moment in time.

    Although I took that understanding for granted, as you emphasized this in your answer (in your Mule/$100 example), maybe you thought I had forgotten that?

    By the way, the same would be true in my Joe and Fred example.

    Isn’t the above what you meant when you said:

    “The key is that while there is no net change in assets/liabilities, there is a change in money as the liability is considered money and the asset (the loan) is not considered ‘negative money’. It is in this sense that people speak of loans ‘creating’ money.”

    Now, let’s assume for simplicity (as things are getting confusing) that there’s only one bank in town, that the loan was made for the term of one month, but it’s extensible and no interests apply. Let’s also assume that the laundry lady does not deposit any coins she might find lying on the sidewalk, or that the merchant will not deposit anything in her account.

    At the end of the month, the laundry lady (who didn’t use the money she had in her bank account) pays the loan in full, using the balance of her “checking account”. The accountant/teller writes two entries in the banks’ book:

    (1) A withdrawal of $5 in the “Laundry Lady checking account” ($5 liabilities disappear).
    (2) A deposit of $5 in the “Laundry Lady loan” ($5 assets disappear).

    So, the laundry lady has no money left in her account; but she owes nothing. The bank’s balance sheet still… balances.

    If, instead of paying the full loan balance, the laundry lady had paid $3 (out of the $5 in her checking account), the accountant/teller would modify those two entries above, replacing “$5” with “$3”: the bank’s book keeps balancing. The laundry lady still has a balance of $2 in her account, and still owes the bank $2.

    Why would the laundry lady take up a loan for $5 and not use it? I don’t know: maybe she was unsure she would be paid by the merchant (her boss) and having some back up cash made her feel better. Maybe she was planning to buy something costing $10 (and wanted to pay it with $5 from the bank and $5 from her wages), but she changed her mind.

    However, whatever her reasons to take up the loan, a loan allows her to use the loan money available in the “checking account”. If it increases her purchase power, it may have effects on her demand of goods (but that is another story).

    On the flip side, it increases her liabilities (and the bank’s assets in the shape of loans!).

    If she repays the bank $5, there is no availability left in her “checking account”, her account balance is $0 (so she can’t spend anything “on account of her account” :o and, consequently, it may diminish her demand). And her liability with the bank disappears (and so does the account “Laundry Lady’s loan”, which is an asset in the bank’s balance sheet, or “book”).

    If she repays the bank only $3, she could still spend the remaining $2. And her availability (i.e. “checking account” balance) is still equal to her liability. Likewise, the bank balance sheet reflects a reduction of $3 in the balance of loans made by the bank.

    Bottom line: any loan re-payment reduces her availability of MONEY (either because she uses cash she has in her wallet, or because she uses money in her checking account [!]).

    I may be mistaken, but isn’t this what you meant when you said: “How can money be destroyed? Basically, through transactions that decrease the size of liabilities that are considered money.”

    Obviously, the preceding does not contemplate the more general (and realistic) case where more banks exist, and people actually deposit in each other’s accounts, and so on and so forth. But as we were already getting dogged down, it seems advisable to see if we can reach an understanding in a simpler position, before adding more details.

    [!] Clearly, if the laundry lady uses cash from her wallet, the transactions the accountant/teller registers are different. He still reduces the “Laundry Lady’s loan” by the amount she paid (reducing a bank’s asset account), but increasing the bank’s cash position (which is also an asset: “cash”). Therefore (and unlike the case when the laundry lady pays with a check from her “checking account”) the bank’s net asset position does not change at all (one increase and one decrease for the same amount); and its liabilities do not change either (the laundry lady paid in cash from her wallet). The bank’s balance sheet still balances, but the total amount of its loans has diminished. Notice that, from the laundry lady’s point of view, however she repays her loan, her broader money availability diminishes: in one case through a reduction of the balance in her checking account, in the other through a reduction of her own cash position.

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