I have a rare set of 1960s magic magazines on Ebay right now and the auction closes later today – details here.  Last week I noticed that one of the magazines has a neat puzzle in it, so it seemed appropriate to use as the basis for the Friday Puzzle….

A friend of mine has £100 in a bank account.  Whenever he takes out money he records how much he has taken out and the balance.  Here are his accounts for last week….

WITHDRAWN                                BALANCE

40                                                      60
30                                                      30
20                                                      10
9                                                        1
1                                                         0

There is one small problem – the WITHDRAWN column adds up to 100, but the BALANCE column adds up to 101 – how can this be the case?

If you have not tried to solve it, have a go now.  For everyone else, the answer (kinda) is after the break….

I thought I would change things around this week and so am inviting everyone to post your answer, and I will give a prize (a sign copy of one of my books) to the person who posts the most elegant and clear way of expressing the solution.  Also, feel free to comment on the solutions!

I have produced an ebook containing 101 of the previous Friday Puzzles! It is called PUZZLED and is available for the Kindle (UK here and USA here) and on the iBookstore (UK here in the USA here). You can try 101 of the puzzles for free here.

1. The balance column does not have to add up to 100. The trick is making it add to 101, people imagine it Should be 100. The trick is there is no trick at all.
Withdrawn Balance
1 99
1 98
98 0

sums are then clearly still 100 for withdrawn but clearly not 100 for balance.

1. Berhard says:

Correct..

but i prefer the simplest solution just withdraw 100 at once..
Withdrawn 100 Balance 0

2. girish says:

Daer Evan
can you,plz explain it in a better and easiest way

2. Mike Hitchcock says:

The balance total does not have to equal the withdrawals total – in fact, it rarely will. Imagine your friend withdrew one dollar at a time….

3. C Hook says:

The ‘Balance’ column is a log of the remaining funds, not a running total of the amount that was present.

To give a more extreme demonstration, imagine withdrawing only £1 each time. After two £1 withdrawals you would be left with a balance of £99 then £98. These two balance values alone add up to £197.

4. Captain says:

The short answer: there’s no reason the BALANCE column even should add up to 100. It can be anything, depending on the withdrawals made. For example if you withdraw the last 10 pounds at once, the balances will add up to exactly 100, but this is just one special case, like 101. Withdraw everything at once, and the sum of balances will be 0, and so on. On the other hand, the WITHDRAWN column will always add up to 100.

5. There’s no reason at all why the two columns _should_ add up to the same thing as they contain completely different views on the data.

This becomes obvious if you add in the missing first row in the table which would read “withdrawn: 0, balance: 100”. It’s then easy to see that there is no relationship between the totals of the two columns.

All that has happened is that you have deliberately chosen a series of withdrawals where the running balances come close to adding up to the same total as the total money withdrawn. The majority of withdrawal patterns would not have exhibited this behaviour. For an extreme example, look at what would have happened if the withdrawals had each been £1.

6. deepak says:

suppose i withdraw 100 at first time , my balance sum will be 0 . !!!!

7. the problem is clearly that we’re using pounds in this puzzle. when we measure data in u.s. dollars, the withdrawal and balance columns will always add to the same number. but the british pound doesn’t function properly within the laws of math.

1. Confmat says:

But they work perfectly fine with the laws of maths.

2. Juan Upman-Schipp says:

We have a winner!

3. Engywuck says:

Tried it with Dollars as well.
Only Euros work!

4. DeepField says:

The problem is not the currency: the bank should not have allowed its customers to withdraw capricious amounts in compete disregard to the laws of decent Mathematics!

5. Holden Caulfield says:

This guy’s won it by a country mile, surely, who could explain it better than that?

6. Anonymous says:

Lol

8. Gib says:

There is one small problem – the WITHDRAWN column adds up to 100, but the BALANCE column adds up to 101 – how can this be the case?

Well, that’s really 2 questions. The first one, which is why the WITHDRAWN adds up to 100, is because 100 was withdrawn.

The second question, about why the BALANCE column adds to 101, is because:
5x – 5a – 4b – 3c – 2d – e = 101
(where x is the starting amount of 100, and a,b,c,d,e are the amounts withdrawn, in order).

That function comes from the first balance value equaling x-a, and the second equaling x-a-b… When you add them all up you get the equation above.

Since you didn’t ask why the sum of the withdrawl amounts and the sum of the balance amounts are NOT equal, or why they are so close, I don’t see any need to comment on either.

1. Anonymous says:

As a matter of fact, if the withdrawals are all whole number amounts of at least 1, the sum of the balances can be anything between 10

Withdrawals Balance

96 4
1 3
1 2
1 1
1 0

and 390

1 99
1 98
1 97
1 96
96 0

2. Jerry says:

Sorry. That should not be Anonymous. It is Jerry.

3. Anonymous says:

9. M says:

The way the question is asked, makes you think the total of both columns have something to do with each other. That is a psychological effect, strengthened by the chosen values. That is because the next value in the right column should be “100”, and the right columns total is “101”. That is just coincidence.

Of course, the total of the left column, should add up to a value in the right column in the next row. But the total of the right column does not make any sence. In this example, you count the “1” four times, the “10” 3 times, the “30” twice and the “60” once. That is because they are already summed in the total.

You can see the problem when you change the 40 into 41, or to any other value.

10. The people above me have pretty much covered it already, but since it’s for a prize, I might as well give it a go.

The answer, simply, is that there’s no particular reason that the Balance column should add up to 100, or even close to it. This can be seen easily through a couple of extreme examples. If the friend withdraws the full £100 at once, the statement will appear as:

WITHDRAWN BALANCE
100……………….0

This will result in the Balance column adding up to 0. At the other extreme, suppose the friend withdraws the money £1 at a time. Then it would look like:

WITHDRAWN BALANCE
1……………………99
1……………………98
1……………………97
1……………………96
(…)
1……………………0

In this case, if you’re familiar with how to “triangle” a number, you can get that the Balance column will add up to 99*(99+1)/2 = 4950. If we assume that money is never deposited, and £1 is the minimum withdrawal, then it’s in fact possible for the Balance column to add up to any integer between 0 and 4950.

The trick to this problem is in making it appear as if, on first glance, the Balance column should add up to 100. There are many possible ways to choose a series of withdrawals that will leave an answer close to, or exactly, 100. For instance, here’s a sequence that results in exactly a sum of 100 in the Balance column:

WITHDRAWN BALANCE
50…………………50
20…………………30
20…………………10
5……………………5
2……………………3
1……………………2
2……………………0

Being close to, but not exactly 100, likely strikes many people as odd, which is what makes this a puzzle in the first place. But in this case, it’s only close to 100 because the sequence of withdrawals is constructed so that the sum of the balances will be 101. The only trick is in convincing the reading that there is, in fact, a trick.

11. Anders says:

The sum of the withdrawals added to the current balance should add up to the initial balance, but there is never a reason for adding old balances together.

Knowing you had 50 pounds on your account on Monday and 40 on Tuesday may be interesting information, but it doesn’t mean you had 90 on Monday and Tuesday added together

12. James says:

The balance sum doesn’t have to add up to 100 it would depend on the amount withdrawn at each stage. The only thing you can can use balance for is with withdrawn so current balance plus amount withdrawn should add up to previous balance.

13. Krishna says:

Since the standard answers already seem to be taken, let me try a different tack:

From http://en.wikipedia.org/wiki/Missing_dollar_riddle:
————————————————————————————-
Three guests check into a hotel room. The clerk says the bill is \$30, so each guest pays \$10. Later the clerk realizes the bill should only be \$25. To rectify this, he gives the bellhop \$5 to return to the guests. On the way to the room, the bellhop realizes that he cannot divide the money equally. As the guests didn’t know the total of the revised bill, the bellhop decides to just give each guest \$1 and keep \$2 for himself.

Now that each of the guests has been given \$1 back, each has paid \$9, bringing the total paid to \$27. The bellhop has \$2. If the guests originally handed over \$30, what happened to the remaining \$1?
————————————————————————————-

It is this remaining \$1 which found its way to the total of balances here so in the end, it all adds up. 😉

It also seems to have become £1 in its trans-atlantic journey without any consideration for currency exchange rates. 😉

14. I was going to have a go at explaining this, but I think the prize should go to Infophile – a lovely explanation which should get extra points for includIng the formula to triangle a number. I don’t think I could do much better than that.

15. jus says:

Why should the balance column add up to the original sum? It just doesn’t.

16. When you add up all your withdrawals, you are counting each pound withdrawn once, and so if you clean out your account over a series of withdrawals, and then add up the sums you withdrew, you’ll get the initial balance (in this case £100).

But if you add up the amounts left each time, you’ll be counting some of the same pounds over and over again. By this process you can reach almost any amount. If you withdraw £10 on day one (leaving £90) and £10 on day two (leaving £80) then adding up the balances gives you 90 + 80 = 170! But you’ve counted some of the same pounds twice, so this is rather a meaningless number.

By arranging withdrawals so that the balances add up to nearly £100, the question attempts to fool you into thinking that they should have added up to exactly £100. But this isn’t true. Think what the sum of all the balances would be if you withdrew £1 a day for 100 days! £99 + £98 + £97…

1. This is a good answer, in my opinion, and better than the ones that just demonstrate that the columns don’t need to add up. I still have a very strong feeling that the columns *should* add up to the same number, and this is the only answer that explains why this illusion occurs. As Jeremy says just below, that’s the important point of the puzzle, not the fact that it’s apparently blindingly obvious to everyone else (and the commenters on Friday’s post) that “there is no puzzle here”.
There are many people like me for whom these puzzles are baffling (I’m still struggling with the hotel bill one, though I now understand the problem).

17. Jeremy says:

The beauty of the puzzle lies in the fact that it is designed so that the totals of the two columns differ only by one. If they were wildly different, as would be true in most cases, it would be more obvious that there is no reason for them to be the same.

I think most people realise this, but just saying ‘there’s no reason why they should add up’ misses the essential point that the puzzle’s specific terms trick the mind into thinking they ought to.

1. Jeremy says:

…in other words, the more interesting question is not ‘why don’t they add up?’, but rather ‘why do they look like they should?’

The short answer is stop getting the public to write your solutions for you. Shameless.

19. “For everyone else, the answer (kinda) is after the break….” except it did not.

1. DeepField says:

But it did, Jus: not from Dr. Wiseman, but from everybody else. 50+ times, so far.

20. Edgar says:

The fallacy inherent in the question arises from the fact that the two amounts measure different things. The balance function is essentially related to the integral of the withdrawal function. There’s simply no reason to assume that one will ever equal the other. The assumption that they should, and the conclusion that the absence of equality requires explanation, exploit most people’s difficulty in understanding accounting concepts and their fear of figures.

21. Mushroom Gordon says:

22. Everyone is trying so hard to post an nice and elegant explanation only because their is a price involved… isn’t that also a psychological effect?

K:)

1. katie k says:

Yes, but only inasmuch as anybody participating in a competition with a set prompt and prize is.

23. katie k says:

The trick of this puzzle is making you believe that the balance ought to add up to 100. A simple example will show why this isn’t so. Imagine you have that same £100 in the bank, but every day you withdraw only £1. After the first day, your balance will be £99. After the second day, your balance will be £98. It quickly becomes obvious that the sums of the various days’ balances are not bound to the original total in the bank the same way the sums of the withdrawals are.

24. Jimbo says:

“How can this be the case?”

It’s very simple: your friend changed a mathematical constant when he took £9 out, so maths stopped working. If he’d stuck with tenders like a normal person instead of going into the branch and bothering the cashier for a single pound coin, none of this would have happened. Never play poker with his guy, or stand near him at a urinal.

1. katie k says:

I never knew my friend was such a jerk. Thanks Jimbo! 🙂

25. Roland says:

Guess this is a puzzle full with white lies and half truths 🙂

1) The balance column simply adds up to 101, because these figures add up to 101. There is no problem at all, it’s fully correct (only the addition has no real meaning), but it is stated as if it shouldn’t. Nothing more, nothing less.

2) The answer is (kinda) after the break, but it is not, at least not given by Richard. Which answer in the replies is the correct one: have a guess 🙂 . Perhaps there is no correct answer…

3) And I now wonder which book will be the prize: “one of my books” can be anything from an old Donald Duck to the bible, depending what Richard has in its book case; while most people probably expect to get a book written by Richard 😉

26. Jimbo says:

*tenners

Darned autocorrect!

27. Paul says:

The columns aren’t supposed to be equal. A good way to show this is to take it to the extreme, so Imagine having £1,000,001 in the account and withdrawing £1,000,000 and then withdrawing the remaining £1, the table would look like this

1,000,000 1
1 0

clearly these aren’t equal or anywhere near equal. The “trick” is only based on the fact that there is a difference of £1 thus making you think that there is a problem.

Or just go into the bank and complain. Most bank managers these days wouldn’t be able to figure it out so they’d probably give you a pound as a refund.

28. First of all, I’d like to note that it’s strange that the initial amount isn’t listed on the balance. The column should start with 100, the total amount would then be 200.

Second, consider another possible representation of what happened:
100 – (a1 + a2 + … + an) = 0,
where 100 is the initial amount and a1, a2, …, an represent the withdrawals. This can be rewritten as
100 = (a1 + a2 + … + an),
which is precisely what happens in your first column and explains why the total sum of withdrawals adds up to 100.

Now consider what the second column looks like in this representation:
(100 – a1) + (100 – (a1 + a2)) + … + (100 – (a1 + a2 + … + an)) = x
In your case x equaled 101 and your implied question was whether this shouldn’t be 100. In fact, all you know about the amounts a1 … an, is that they add up to 100. That leaves many possible values for x, ranging from 100 (one withdrawal of 100) to 4950 (one hundred withdrawals of 1), or even more if you allow withdrawals of fractions.

So, the way you asked the question suggests that x should be equal to the initial amount on the balance, but in fact its value is only related to that amount in that it should be greater than or equal to it.

29. Nick says:

It’s hard to add much to everyone’s comment above, so I will just note that either column — “withdrawn” or “balance” — completely determines the other. The balance column can be a completely arbitrary sequence of numbers ending in zero, the only restrictions being that if the friend makes only withdrawals the sequence of balances must be non-decreasing and the first term must be less than 100.

Therefore to construct a sequence of withdrawals whose corresponding balances sum to, say, 100, simply write down the balances — e.g., 100 > a > b > c > d > 0 with a + b + c + d = 100 is a list of four balances (a, b, c, d) — and calculate the corresponding withdrawals as: 100 – a, a – b, b – c, c – d, d.

For example, a = 50, b = 25, c = 15, d = 10, generates:

WITHDRAWN BALANCE
50 50
25 25
10 15
5 10
10 0

Now both columns sum to 100!

30. Lazy T says:

Suggestion + Misdirection = Fallacy

1. OK, this is definitely the most elegant condensation of all of the above.

31. spamsac says:

There is no more reason for the sums of the columns to be equal than for the equation “apples=oranges” to be true: they’re different things.

32. jpr says:

the most elegant and clear way of expressing the solution. Also, feel free to comment on the solutions!

Exactly as requested

Richard said “and I will give a prize (a sign copy of one of my books) to the person who posts the most elegant…”

Some of you just don’t think outside of the box!

33. Bletherskite says:

“How can this be the case?” – because it is.

Good day.

34. safc4ever says:

As the balance is a result based on the accumulated total of withdrawals subtracted from the original balance, it cannot be accumulated itself to give any useful data. The fact that you have done so and found that the result is only 1 away from the accumulated withdrawals is irrelevant.

1. safc4ever says:

I could have used ‘unrelated instead of or in addition to ‘irrelevant’.

35. Julia says:

I don’t seem to be able to post replies to individual answers any more, so I’ll try here:

Katie K said exactly what I would have said, but I like the way Anders explains things too – simple and effective.

1. katie k says:

Thanks Julia! I’m glad we’re on the same page. 🙂

36. neil jackson says:

You don’t total balance columns. USE YOUR BRAIN NUMBNUT.

1. neil jackson says:

Too harsh?

2. Harsh maybe, but when you put it that way, the solution is very obvious. Good answer.

37. The sum in the “Balance” column shouldn’t be the total amount of money he had, because the same money are counted more times (for example, the final pound will be taken into consideration in all the previous “Balance” values).

You get the total £100 by adding the “Balance” value on row x and all the “Withdrawn” values from row 1 to row x.

38. Matt says:

Aw Richard it’s not your day is it?
First you post a terrible ‘puzzle’ then you misplace the ‘answer’ and now you’re stuck with the impossible task of judging dozens of identical ‘solutions’.

39. chris oddo says:

The sum of the two columns will very rarely add up to the same number. If you have 100.00 dollars the withdrawal column is hallways going to eventually add up to 100.00 dollars.but if you withdraw 1 or two dollars at a time the added sum of the balance column would be extremely higher than the withdrawal column. Very simple there’s no trick here at all.unless your a simple mind.LOL

40. bigtourist says:

the balance is a list of amount over time

the withdrawal list is the rate of change.

if you were to plot this out as a graph, the balance column would define the Y parameters across time X. The withdrawal list is just a way to be defining the curvature of the line through those points.

Both columns are simply 2 distinct ways of expressing the same graph. Any instance where the numbers did sum equally would be pure coincidence.

41. There is no reason whatever for the customer’s original deposit of £100 to equal the total of the balances left after each withdrawal. The total of withdrawals in the left-hand column must always equal £100, but it is purely coincidental that the total of the right-hand column is close to £100.

42. Per says:

This is really banking 101 🙂

43. nanda says:

Lets see who is going to get a book.

44. fredb says:

The sum of the balances is irrelevant. Imagine if each week you withdrew £0 – your balance would be £100 each week, giving you a very large sum after several weeks.

45. Adding up the individual withdrawals will provide the total withdrawals.
Adding up the running balances has got nothing to do with the price of fish.
Simples.

46. DeepField says:

If the above solutions have yet to convince somebody, here is mine: imagine you had a way to identify each of the initial £100, a different name or a different color for each of them. The first day you withdraw your least-favorite £40, and count the remaining £60 on the right side column.

The next day you go and get the following £30, and have a look at your remaining £30. You should have realized that you had already counted them the day before, and that adding them to the running total, though a valid operation if you want, for example, to know your average balance, does not have to match the initial balance you had.

47. A given pound will be counted only once in the WITHDRAWN column, that being when it is actually withdrawn from the bank. The same does not hold true for the BALANCE column. The last pound withdrawn, for example, will be counted among the balance upon every withdrawal.

48. Natalie says:

The balance is the total at that particular time and although varied by the withdrawals it is only ever the one amount shown. The balance column should not be totalled up as the previous amounts no longer exist. Each is replaced by the next.

49. The sum of your balance doesn’t have to be (and rarely is probability wise. It is usually more.) the same as the amount you withdraw. Imagine you have £100 (or any other currency, to please some people commenting…), and you take out £1 at a time:

W – B
1 – 99
1 – 98
1 – 97
1 – 96… etc.

Quite clearly the sum of the withdrawn will be £100 (if you take out all the money) but also quite clearly, the sum of the balance will be more than £100 (much much more in this example).

The part where this riddle confuses people is that the sum of the balance is very close to the sum of the withdrawn and this fact usually catches people out.

Summing up your balance remaining is pretty pointless as this shows.

1. Irrelevant would be a better word than Pointless.

50. Ed says:

Imagine the bank account as a bag of 100 actual pound coins, and imagine you can tell them all apart.

Imagine that Richard’s table contains not just numbers but lists of the exact coins that are being withdrawn or left in the bag.

Each coin appears exactly once in the withdrawals column, so that column has 100 coins in total.

A coin MIGHT appear once in the balance column. But it could appear several times (if it’s left in there for a few withdrawals), or not at all (if it is pulled out straight away). So we just can’t tell how many coins will appear in this column.

51. JamesMaullin says:

The balance column doesnt need to add up too 100.

Imagine you have an account with £100 in.

Now imagine you remove a pound from this account every day for 100 days. After two days the table would look like this:

WITHDRAWN BALANCE
1 99
1 98

Now looking at this it is clear the total for the balance column is already more than 100 and it is also clear that it doesnt have to do so.

If you need to get hold of me my email is:

Mechscapefan@live.co.uk

52. James Maullin says:

Apologies that the columns dont line up – either my iphone or this website corrected it even after i left spaces.

53. SimonP says:

Human’s are very good at finding patterns and relations even where they don’t exist. When we are given two similar answers we assume that both have similar causes. This completely blinds us to the real reason for the similarity of the two answers – in this case the devious mind of the puzzle setter. This same effect is seen at work when we perceive conspiracies, ghosts, alternative medicine and many other forms of ‘Woo’.

Very simple.

It can be explained with Giggs Boson theory.

It doesn’t matter how many injunctions I take out (my withdrawals), there will be always be a higher number of people who know about my deposits.

55. Who cares about the solution — what about your friend? He’s dead broke! Will you spot him a fiver from your eBay booty for some tea and crumpets (or whatever you Brits eat nowadays)?

56. safc4ever says:

Let us consider another example: You start with 1 of each UK coin in your purse and spend them in decreasing order of value. You start with £2.00+£1.00+£0.50+£0.20+£0.10+£0.05+£0.02+£0.01 = £3.88

The total spent will be £3.88; If you add the remaining balances, you are adding the 1p (£0.01) 7 times, the 2p 6 times and so on up to the £1.00 once. The £2.00 coin never appears in the balance as it is spent first, before the first balance is taken. The balances will be £1.88, £0.88, £0.38, £0.18, £0.08, £0.03, £0.01 giving a total of all balances of £3.44!

Of course, this bears no resemblance to the original £3.88, but that’s because balances on a single account should NEVER be added. You can add the balance of what is in your purse with what is in your hand at the same moment, but NEVER add what is in your purse one day to what is in the same purse the next day.

1. safc4ever says:

In case there are any pedants who have a ‘Crown’ (25p) or £5.00 coin in their purse, these value coins have only been issued as commemorative coins and are not in wide circulation. They are, of course, still valid coins but not enough exist for me to include them in the above example.

57. Gabriel says:

Take {xi} to be the “withdrawn” sequence in reverse order: e.g. {xi}={1,9,20,30,40,…}, and {yi} the “balance” sequence in reverse order: {yi}={0,1,10,30,60,…}. Then by definition y_i=sum_{k=0}^{i} x_k
But this in no way implies sum_{k=0}^{i} y_k = sum_{k=0}^{i} x_k

58. Paul says:

The appropriate equation is: balance + withdrawal = previous balance
Not: balance + previous balance = withdrawal + previous withdrawal

59. Ersagun says:

If you have a bag and some balls in it and you think you are counting the balls in the bag by counting the balls left in the bag(instead of counting the ones you just took out) every time you take some of them out, it means that you are counting some of them more than once and making me explain THE simplest mathematical concept(counting).

60. Jason says:

I’m gonna plead the case of rounding – if you round down enough, you could add up to that extra 1 hanging out… make sense?

61. Joe says:

The balance column, well, how could it always add up to £100? Supposing you were to get carried away, and go extremely overdue…

Withdrawn____Balance
___100________0___
__1,000______-1,000

e.t.c.

62. Alessandro I. says:

WITHDRAW column works by sum. BALANCE column is the result of an operation of substraction, 100-WITHDRAW available. Nonsense to sum each balance column.

63. The balance is a time critical figure. We forget that the balance is always current and thus has a measure of (money of this specific time period).Thus, adding it up produces a number that has no meaning because of the changing time significance of the figures added. It’s literately adding apples and oranges.

64. Tinfoil Hat says:

The missing £1 was secretly diverted to some shady government fund.

1. Tinfoil Hat says:

By the way, how many puzzles are in Richard’s new book again?

And say the first withdrawal is W1
The balance B1 is A – W1

After second withdrawal W2, balance B2 is B1 – W2 = A – (W1 + W2)
After third withdrawal, the balance B3 = A – (W1 + W2 + W3)
…..
After nth withdrawal, balance Bn = A- (W1+W2+W3+…Wn) in general
Say you make total N such withdrawals to finally make your account empty.
Then after Nth withdrawal, the balance will be
BN = A – (W1+W2+W3+…+WN) = 0 —-equation 1

total = NxA – NxW1 – (N-1)xW2 – (N-2)xW3….-(N+1-n)xWn….- WN —equation 2

(note that general subtraction term for nth withdrawal (N+1-n)xWn is valid for all n from 1 to N)

but using equation 1, we can cancel A with one one each of W1 to Wn.

total of balances = (N-1)A – (N-1)W1 – (N-2)W2 – (N-3)W3…-(N-n)Wn… -0 xWN —-equation 2.

This equation tells us all the magic of total of balances (ok I am talking losely when I say ‘all the magic’. Bet real mathematicians would be able to tell you a whole lot more from the equation!) :

First term – (N-1)A:
1) The more your initial amount A, the more is the total.
2) You can bloat from that amount A as much as you like, by being stingy and withdrawing less and less amount at a time, so that your total withdrawals N is huge!

Term for nth withdrawal (N-n)Wn
1) The smaller amounts Wn you withdraw, the smaller the term (N-n)Wn will be. So less will be subtracted from first term and your balance column total will be larger. This complements point 2) of first term above. By withdrawing less Wn at any given time, you increase the total withdrawals N needed. Both of these increase your balance column total!
2)The larger your initial withdrawals (for small n), lesser will be your balance column total, since that amount multiplied by (N-n).So if you have decided the total number of times N you will withdraw, you can still make your balance column total larger by being stingy initially!

To give a roundup conclusion,
1)The minimum balance column total you can have is 0, by making a single, one sweep withdrawal of the whole amount A. Though this is quite obvious, if we want formality,
N= 1,
For Wn = A for n = 1
Wn = 0 for all other n
So, balance total = (1-1)A – (1-1)A = 0
2)The maximum balance column total can go to infinity. When each withdrawn amount Wn approaches 0, the total withdrawals N you need to make will approach infinity!
So balance total = (N-1)(A-W1) – (N-2)W2 -…(N-n)Wn…
Hmm ok, here we have N becoming infinite and Wn near zero, so you might question the product from second term onwards, whether it is ‘undefined’ or 0. My gut feeling says the total balance could become infinite, because say you start with 100, and after withdrawing tiny (infinitecimal) amount, you still have balance 100 every time. So we can add it infinite times and get an infinite total.

But that’s all for now folks, got to go! Might come back later with proof that the total can indeed go to infinity by withdarwing infinitecimal amount from account each time (of course spoilsport goverments will hamper your noble mathematical endeavor by issuing only a certain minimum denomination loose change :P, and maybe made even worse by your bank’s minimum transaction limit! Not to forget that based on your country laws and bank rules, you might be charged a non infinitecimal fee from the same account, just for processing your withdrawal! :P)

66. Vern says:

Realise that the Balance column would feature 100 before any amount was withdrawn. This will shatter the illusion that both columns should tally 100.

WITHDRAWN BALANCE
0 100
40 60
30 30
20 10
9 1
1 0
——————————
100 201

Why don’t they tally? Because Balance columns are a running remainder. Take all your money out in one hit and they’ll balance. Any other withdrawal you make and the balance total will bloat. THE MORE withdrawals you make the more remainders will appear in the Balance column and the higher the total of the balance column will go!

Imagine instead we are counting down to the launch of a rocket. 10… 9… 8… 7… 6… 5… 4… 3… 2… 1… Would it make any sense to ask why a ten second countdown doesn’t take 55 seconds?

67. There is elegance in simplicity. I only need to know how much I have withdrawn so my impulse buying can continue. Of course, since I deal with my own bank account in this manner, I find that I have to stay employed to keep that “invisible” balance column above zero. Ce la vie!

68. Dwatney says:

If you think the sum of withdrawals should equal the sum of balances, you have a great career head of you in government!

69. Noel says:

\$100 – \$40 = \$59

70. Anonymous says:

Simply because adding up the WITHDRAWN column includes each pound once, while adding up the BALANCE column includes the same dollars multiple times.

1. CH says:

whoops, that was CH. And sorry about mixing pounds and dollars, I’m a Texan trying to speak “English”

71. Andrew MW says:

There was a remainder called “Balance”
Whose failure to sum was a nuisance.
He replied “I wouldn’t!
I’m what’s left, not what’s taken, in essence.”

1. Andrew MW says:

Apologies, there is a scansion error in 4th line – please change “replied” to “stated”, thus:

There was a remainder called “Balance”
Whose failure to sum was a nuisance.
He stated “I wouldn’t!
I’m what’s left, not what’s taken, in essence.”

72. Chris Emerson says:

Summing the withdrawn column counts each pound only once, and eventually (when the account is empty) equal the amount of money that was in the account in the first place – 100. Summing the balance column counts some of the pounds (the last ones to be withdrawn) several times.

The values in the puzzle are constructed in a way that the balance column adds up to something very similar to the amount withdrawn column, thus creating the feeling in the reader’s mind that they should be equal, and that only a small error has happened, when in fact the 2 sums have absolutely no reason to be close to each other.

The addition of the initial balance of 100 pushes the sum to 201, destroying this notion that the values should be close. 40 of the balance is never counted at all (withdrawn before the first value in the balance column). Since each value in the balance column is already included in the sum of the previous value, 30 is counted once only (second value subtracted from the first value), 20 is counted twice (2nd – 3rd), 9 3 times and 1 4 times.

30 + 20 + 20 + 9 + 9 + 9 + 1 + 1 + 1 + 1 = 101

73. JR says:

The balance column does not need to sum 100. The amounts for this example were selected to trick the public, but simply imagine if you withdraw 1 pound at a time, here the balance column will sum more that 100 from the first 2 withdrawals.

74. The 1 is part of the 10, the 10 part of the 30, the 30 part of the 60, and the 60 part of the implied 100. Leaving the beginning balance out creates the illusion that the list of balances should add up to 100, but you would never add the 1 slice of pie you’ve eaten to the original 8 and wonder where the 9th piece came from.

75. Ian says:

The total of the withdrawn amounts at any stage plus the corresponding balance must always equal the original balance, however whilst the withdrawn amounts are cumative, a balance is a snapshot in time and therefore not intended to be cumulative.

76. SereneBlueSky7 says:

In the balance the final 1 is from the previous 10 and gets counted twice. 1 repeated two times, adding a pseudo 1 to it all. 🙂

77. joshthelegend says:

i think i’ve got two friday puzzles right before. this wasnt one.

78. rmb says:

So who is the winner?

79. sameer dongre says:

since WITHDRAWN and BALANCE are interdependent, cummulative addition is applicable here and not the regular addition of the indivisuals (bal and withdrawn amnt.)

The only reason of the difference in sum of both columns is that ……in 1st column we r counting from 1 to 100 and in 2nd column its going from 100 to 1. so if we count 3 in 1st column it will give a value of 3 while in 2nd column it will give 97. in this way the smaller counts wil create bigger difference while bigger ones will reflect little differences.

for the hotel bill puzzle….the answer is that the amount paid by the guests includes the amount charged by the clerk(25) and the amount taken by bellhop(2)=27 and remaining 3 were returned to them so the total counts to 30

81. This puzzle is useless and time wasting because there is no meaning in counting the what has left. If you go to store and buy something. you would only count money by what you have spent not with what you have left.

82. Emily says:

How much money is subtracted from the balance column

83. Kelvin says:

It is simple it is because you can’t withdraw £1 or £9 from the bank they don’t give coins

84. What are real products of 108 and if you add them they become 33?

85. LOKESH KUMAR says:

Instead of withdrawing amounts in this sequence 40, 30, 20, 9 & 1, If you had withdrawn in the sequence of 40, 30, 20 and 10, total withdrawn and total balance amount would have been same. This is the only way to match both the columns equal as a total.

WITHDRAWN BALANCE

40 60
30 30
20 10
9 1
1 0
total 100 total 101

WITHDRAWN BALANCE

40 60
30 30
20 10
10 0
total 100 total 100

86. Sam Iyengar says:

Actually here is the general solution to this problem. If you start with an amount Q and make n withdrawals (n > 2) then the Sum of the balances – Q = sigma i=3 to n (i-2) x n(i) – n1. In other words for 5 withdrawals n1 through n5 you can always balance the 2 columns provided the following holds:

n3 + 2 X n4 + 3 X n5 = n1

For 10 withdrawals from 300 try these numbers:

220,20,10,10,10,10,8,6,4,2

87. Andy says:

you can only withdrawal what you have, that is 100

88. Howale says:

1 is used twice so by counting 100+1=101

1. Howale says:

So 1is extra

89. kavita says:

101-100=1
Actually, left out balance ko hum matter krnge to 101 hi aayega. Iss lye hume issko mind me nhi lena chahye and 1 rupee balance carry forward ho jayega
So, 100-(40+30+20+9+1)
100-100
=0

90. pawara naresh says:

100 -40=60
60-30=30
30-20=10
10-10=0
—————————-
100= 100