Another of my favourite puzzles this week. Imagine that you accept a new job and are offered one of two possible pay schemes.

Scheme 1) You get a starting salary of \$100 per year, and then a \$20 pay rise each year.

Scheme 2) You get a starting salary of \$100 per year, and then a \$5 pay rise every six months.

In either case, you are paid every 6 months. Which scheme is better?

As ever, please do NOT post your answers, but do say if you have a solution and how long it took. I will post the solution on Monday.

Oh, and if you enjoy the Friday Puzzle, remember that I have produced a kindle ebook containing 101 of the previous Friday Puzzles! It is called PUZZLED and is available in the UK here and USA here.

1. JayCee says:

both these schemes are not an option, i’ll have to look for a better paid job then.
or,…. are we travelling back in time, where the dollar was worth something?

10 sec to decide this (40 sec to solve the puzzle)

1. Anonymous says:

Did it in my head. Took 30 seconds. May be right. Could be wrong. Pen and paper – or Monday – will tell.

2. B. says:

The wording of this puzzle is atrocious.

Is your pay \$100 per year, and you are paid \$50 every six months?

What is the interest rate? Or the rate of your time preference?

1. Jazzer says:

Same question here. Is the salary for a week, month, half year, year?

2. Martin says:

Nice one.
Its a no-brainer, took about 2 mins with a piece of paper but could have done it without.
Its really easy and ALL the info is supplied and correct. You just have to READ it and think about it…..

3. Nic says:

It says per year.

3. Morgan says:

To accurately respond, one needs further data, imcluding how long you expect to keep your job, predicted inflation, personal cost of capital…
Assuming you intend staying there indefinitely & inflation/capital cost nil then v easy to work out.

1. Nick says:

In my solution (2 minutes), by the end of the first year, one option clearly emerges as a better choice. You don’t need to stay there forever to reap the benefits of the better option, and inflation/other costs make no difference to the one I’d pick (unless it pushed me over a tax bracket!).

4. It took me a couple of minutes.

I think I must have been a bit confused by the juxtaposition of the dollar sign alongside “pay rise”—much like I would be if the words “pay raise” were juxtaposed with £. Ha!

5. So is that \$100 per year? Or per six months? Either way this seems far too easy. In fact, the answer to this question makes a big difference. If it’s \$100 per month paid every six months then it’s a bit trickier – but then being paid every six months would have no point.

In short, am I missing something, is the puzzle missing something, or is it just way too easy?

6. I made the assumption that it was an annual salary (i.e. you are starting at \$100 for the first year) and interest was not included.
The answer I got does not depend on how long you stay there. However I made a couple of more assumptions that I can’t go into without revealing too much of the answer.

7. M says:

Seems to easy. What am I missing?

8. Englishdood says:

an “o”…

1. Carogmatic says:

Best response ever.

2. M says:

thanks… that helps…

3. Chimera says:

Win!!! 🙂

9. I think it depends a lot on whether we’re living check-to-check or if we’re able to deposit into a high-interest savings account.

10. Matt says:

I think the trick has to do with when you earn vs. when you’re paid.

11. Mick says:

got it, 5 minutes and with Exel

12. Ten seconds to confirm my initial instinct on a piece of scrap paper.

13. Mark says:

I’m rubbish at maths so either this was too easy – or I’ve got it hoplessly wrong! I didn’t even need a bit of paper to convince myself of my answer.

It’ll be interesting to see the real answer next week. 😉

14. Mchl says:

Some 15 seconds to realize what is wrong with the puzzle. I will eagerly await monday! 😀

15. Anonymous says:

It depends on how you interpret the \$5 pay rise. If you interpret it as a \$5 rise in the 6-monthly payment, the answer is one thing, if you interpret it as a \$5 rise in annual salary, the answer is another thing.

1. Emlyn says:

Yes, I just worked this out too! I suppose it is meant to be interpreted as a \$5 rise in the 6-monthly salary, otherwise it is not really a puzzle, but I don’t think that is the correct interpretation as worded. Scheme 2 should have been a starting salary of \$50 per 6 months, then it would make sense.

Or maybe I’m just missing something…

16. Timdifano says:

A pretty clear answer in a couple of minutes, needed pen and paper though.

1. Hubert says:

He’s planning for the future.

17. David Mathew says:

Suspiciously easy. I’ll look forward to being proved wrong on Monday 🙂

18. JimC says:

I had a hunch when I finished reading the question, and a minute or two with a piece of paper confirmed my suspicions.

19. neil j says:

I have not one answer but 2, depending on a slight ambiguity in the question.

1. nanda says:

yes. Depend how old you are?. How long you are going to work

20. Berber Anna says:

I don’t get whether the rise is in addition to the annual salary or the semi-annual one that you’re actually payed.

1. sc says:

I think that, as written, you can only take this to mean a raise onto the annual salary, though I’m guessing this may not be what was intended.

21. somesh says:

22. slw says:

I don’t get it, the answer seems entirely obvious, why is this a puzzle?

23. I’m hoping that the answer is more interesting than the obvious solution. To accept one one the offers one would not be a very wise-man.

24. Anataboga says:

This doesn’t seem to warrant being called a “puzzle”. It’s just a simple maths question.

1. agree – and to make it more topical it could have used £100,000 salaries, and £200 a year annual increases, versus £50 6 monthly,
and even more fun in maths: – percentage compound increases and a varying interest rate – but hey
its for the masses…

25. Rob says:

Neat. Solved by simple brute force in a few seconds, then a minute or two spent trying to ‘intuitise’. Got it now.

26. Great stuff, I knew S=1/2n(2a+(n-1)d) would come in handy at some point

27. Joao Pedro Afonso says:

done in 30 secs/1 min. I like this puzzle very much although to called it a puzzle leads automatically to the solution (twisted human logic shows which should be)

28. Goliath says:

Not sure if this is a puzzle at all. Put it in an excel sheet and got it “solved” in a few clicks. It’s really just putting in the numbers given in the text. ;(

29. mittfh says:

I got what seems to be the obvious answer within a few seconds of reading it, then spent another minute re-reading the question to see if I’d missed something obvious that would point to the other answer.

I couldn’t, but would Richard really set a Primary School maths problem?

I’m just loving all the ‘this is not a puzzle’ comments. Can’t wait until Monday 🙂

31. David Mathew says:

It’s looking less and less likely that my ‘solved-it-in-a-minute’ answer is going to be correct. Still, ’twas ever thus – and all that – with me at any rate. 🙂

32. Dean says:

I think iv got this one, will wait till monday to see

33. ivan says:

The puzzle only makes sense (as a puzzle) if the \$5 pay rise means a \$5 increase in the 6-monthly rate, ie, he would receive \$55 for the second 6 months under that option (rather than \$52.50). So I’m guessing that is what Richard meant, even though he doesn’t really seem to have said it.

1. sc says:

It doesn’t really make any less or any more sense either way, just affects the outcome. However, there doesn’t seem to be any justification for handling the pay raises for either of them differently.

Ignore the fact that the pay raises are coming regularly and just take an example where the boss says I am giving you a pay raise of \$20 to one and \$5 to the other. They are both paid every six months so you either have to attribute this to a raise in the money they take home each six months or the money they take home each year…in both cases. With identical wording there is no reason to assume anything else.

So if you are assuming the \$5 pay raise means an increase in the 6-month rate for one employee you have to assume the \$20 pay raise for the other also means a raise in their 6-month rate.

(Note: I don’t think this was what was intended, though I can’t see how there is room to interpret this in any other way).

2. Riiick says:

Assuming an annual rise, a \$9.70 rise for the second option would make things more interesting (\$4.85 every 6 months).

3. ivan says:

Clearly there is “an answer” whatever the correct interpretation is. The question is, is it interesting enough we can call it a “puzzle” rather than an easy arithmetical exercise.?

It seems to me that, to be within the meaning of the word usually given to “puzzle”, then something like one of the following has to apply.
(1) The option that, at first glance, looks lower is actually higher,
(2) The options look different at first glance, but turn out to be the same
(3) The options look the same at first glance, but one turns out higher than the other

The difficulty with the literal interpretation is that there is no puzzle factor in the answer, it’s just a trivial arithmetical exercise. One option looks a lot higher than the other because it is a lot higher than the other, a few quibbles about hyperinflation and contract length apart.

I therefore interpreted it in a way that there was actually a “puzzle factor”.

4. underblog says:

Agree. I initially assume “\$5 pay rise every 6 months”, meant “\$5 per annum pay rise every six months” which I guess doesn’t lead to the intended result. All the other numbers are pa figures and that is the conventional practice so wording could be a bit more explicit.

34. CH23 says:

This one’s quite easy too(or that’s what i think)

35. Nic says:

So many pedantic people commenting here. Stop trying to trick yourself with inflation issues, interest rates, etc. Just make some simple assumptions and work it out that way. As long as your result is correct according to your assumptions then you’re fine.

Took me about 2 minutes to confirm.

36. It’s not as obvious as you might think…and you do not need interest rates etc. to come up with the right answer.

Hint: think little and big steps.

@Ivan – I think you’re on the right track.

37. I admit defeat. I do not see a puzzle. And I’ve read the words over and over and over and over again. The thing I miss must be too obvious to be seen.

…except the fact that I would accept *either* salary myself 😉

/Z

1. Make that “wouldn’t”, of course. *d’oh*

/Z

38. Bletherskite says:

I got my answer in about a minute working it out on a bit of paper.

39. Francis says:

Usually, we have a “salary rise”, which is based on our yearly salary. But in this case, if the “pay rise” means an increment on each pay, it changes everything!

1. sc says:

It actually doesn’t because you would have to assume that for both of them, and they are both paid every six months 🙂 It only makes a difference if you interpret it as “each pay” for one, and annual salary for the other. Of course there is no reason given that indicates they should be treated differently. My guess is that the intention of the puzzle is that the \$5 one is a per-pay increase and the £20 a yearly salary income, though that isn’t the puzzle we have been given 🙂

40. Grant says:

I wish everybody would pipe down about the request for definitions and interest rates and wordings! Unless I’m completely wrong, the answer is very simple and took less than a minute.

Looking forward to the pro rata argument on Monday..

42. Emlyn and Ivan have to be right. If this is supposed to be puzzling then there needs to something counterintuitive about it. That’s only the case if you interpret it as follows:

Option A is a yearly salary of £100 paid every six months, with a yearly increase of £20.

Option B is a yearly salary of £100 paid every six months, with a six-monthly increase of £5.

That’s the only interpretation which allows it to fit the puzzle model. You can choose another if you wish, but I bet this is the intended one.

43. Anders says:

Yeah, whats up with interest rents etc . This is a PUZZLE people, not an actual offer. You are paid 100\$ a year.You get paid every six months. you can either get 5 more every six months or 20 more every year. period. its not meant to take into account anything but the specified parameters. It could just as well have said “I’m gonna give you 100 apples every year….” Stop overthinking it.

44. Garrett says:

As other people have hit on already, there isn’t enough information here.

The way the question is written the answer is very obvious even at first glance, however I imagine that the question was supposed to be worded differently.

However, even if we re-word the question as:

1) \$100 annual salary, with a \$20 annual salary pay raise once every year
2) \$100 annual salary, with a \$5 pay raise per paycheck

the question is still ambiguous, as changing what time values they get their first paychecks, and when #1 gets their raise can give you different answers.

45. There is all the information you need. Seems like a lot of people are over thinking this one!

Easy easy easy

\o/ /o\ \o/ /o\ \o/ /o\

1. Emlyn says:

Yes as it’s worded it’s “easy easy easy”, so much so that it’s not really a puzzle. That’s why people are “over thinking” it, because it was probably meant to be worded differently.

In fact maybe that’s the whole point, the obvious answer is right – it’s a double bluff puzzle 😉

46. NoAstronomer says:

Firstly I agree that this isn’t a ‘puzzle’, it’s a math problem. I too used a pen & paper and I found that the answer to one of the questions posed by the commenters before me makes a difference. You do have to make some assumptions about when the pay is delivered. Took me about 10 minutes to work through some scenarios.

47. Maybe I’m just too tired but it looks totally trivial to me. As a maths teacher I knew what I was looking for but there was £ in each option and not % in one of them so both APs not an AP and a GP

48. I saw this puzzle (or a close variant) in a really good puzzle book I had growing up.

Can’t remember the name of it, but it had a green cover with some yellow matchstick puzzles on the front, and was standard paperback size — I think it was published by either Penguin or Piccolo. It had some of the most fiendishly complex puzzles my 12-year-old brain had ever seen. I loaned it to my maths teacher, and never saw it again, more’s the pity.

Anyway: I remember the answer being completely counter-intuitive then, and the puzzle has stuck in my mind ever since. I could never remember how their answer was arrived at, so I was delighted to see Richard’s version, thinking that finally, I can get some closure.

Having now read it through, I also think it’s either too ambiguous or else misstated so that it’s not the intended puzzle but a far easier one instead.

In any case, I await Monday’s answer with interest…

1. After reading some of the follow-up links referenced in Monday’s comments discussing the (controversial) answer, I think the book I’m looking for is “Fun with Puzzles” by Joseph Leeming (1946).

In particular, the sample puzzles it contains, described at http://www.donaldsauter.com/brain-teasers.htm, match exactly with what I remember.

Only thing different is the cover, but I expect what I had was a UK edition.

49. Henry says:

Mr. Wiseguy must have edited the puzzle after first posting it because the e-mail alert I received said “a salary of \$100” but now the blog says “a salary of \$100 per year.” Without “per year” included, the wording implied that it was a semiannual salary (“you are paid every 6 months”). It’s one answer if the salary is annual, and the other answer if the salary is semiannual, so I’m happy Richard fixed it.

50. Ive decided I’m not intelligent enough to solve these puzzles, so I read it and plumped for the option that looks worse, on the basis that it must somehow work out to be the best. That took me about 1 second.

1. Jerry says:

I used the same reasoning, then checked it on a spreadsheet.

51. Zach says:

2 seconds, though I’ve seen a variation on this before.

Also, it depends on how long you plan on staying at the job.

52. Jeremy says:

I think this is an excellent puzzle. I guessed at which option ‘looked’ right, then did the maths and found that my intuition was wrong.

It didn’t take long, but was fascinating and surprising none the less.

I can only imagine that those who are arguing that it’s not a puzzle have either got the wrong answer or a peculiar idea of what a puzzle should be.

1. Emlyn says:

I think you have misread the puzzle, and are assuming that with scheme 1 the annual salary increases by \$20, but in scheme 2 the semi-annual salary increases by \$5, in which case it would be a puzzle.
But if you read carefully, there is nothing to say the two schemes should be treated differently, you can argue that a pay rise should be per year, or per paycheck, but not that it’s one in one case and the other in the other case.

53. Jeremy says:

No, I don’t think so. As so often with Richard’s puzzles, I think people try to find ambiguities that really aren’t there. The puzzle is worded perfectly clearly.

If your employer told you you’d be getting a \$5 pay rise every six months, but then only increased your annual salary by \$5 every year, I’m pretty sure you’d be telling them they’d promised something rather different.

1. Francis says:

Yes, it is now because Richard changed the wording. The first version was not specifying when the pay rises occur. Was it yearly or every six months? It was not specified.

2. Emlyn says:

I don’t agree. While I think it is worded quite clearly, I don’t interpret the same way as you.
Salary is counted either per year, or per payment, depending on convention.
If it was a one-off pay rise of \$5 would it be \$5 per 6 months, or \$5 per year? I would interpret it as \$5 per year. Just because there are more pay rises, that happen to be every 6 months, why should the meaning suddenly change?
And if you interpret it as \$5 per payment, then you should also interpret the first scheme as \$20 per payment, changing every year.

Why would you interpret a \$5 pay rise every six months as an annual salary increase of \$5 every year? It should be an annual salary increase of \$5 every 6 months.
Or you could interpret it as a 6 monthly salary increase of \$5 every 6 months, but then you should also interpret the first scheme as a 6 monthly salary increase of \$20, every year.

54. Jcrux says:

Better wording:
Scheme 1) You get a salary of \$100 per year, plus a \$20 pay rise each year.

Scheme 2) You get a salary of \$100 per year, plus a \$5 pay rise every six months.

In either case, you are paid every 6 months. Which scheme is better?

The original wording, which used “starting salary” (and “and then” instead of “plus”) made it ambiguous whether Scheme 2 receives a pay rise during the first year. (A \$100 starting salary implies to many people that \$100 will be the salary for the first year). Suddenly this becomes something more like a puzzle…

I had to look up the puzzle at another site to see what was meant; unfortunately reading the solution was the only way for me to understand the question correctly this week…

55. Tyro says:

I’m totally confused since this seems waaaaay too simple. People are talking about pencils or spreadsheets but to calculate what? I don’t get it. Maybe if one or both were percentages but a fixed \$5/6mo and \$20/yr, I just don’t see what the puzzle is.

Can anyone help make this harder for me?

1. Jeremy says:

That rather depends on what you think the answer is… although I guess we need to wait till Monday, in order not to spoil it.

As I said a little earlier, I thought the answer was obvious, but found, after writing it down and doing some maths, that I was mistaken.

56. Will says:

I suspect a fraudulent answer that will compare apples to oranges.

57. Jeremy says:

I sometimes wonder why some of you do these puzzles… half the people on here seem only to enjoy moaning about the answers or the wording or whatever. It’s not compulsory, you know….

1. Jerry says:

It’s a great day in the moaning. (:-))

58. I used a spreadsheet to work this out – just because I could. The spreadsheet was on a Mapple and I was used to a computer so it took a little longer. Otherwise – very interesting result.

59. Square Image says:

we did in my pre cal class so I found this pretty easy. good one 😀

60. christine says:

Fun Fun Fun

First answer in twenty seconds but it was wrong … ha

Second answer three minutes .. not sure if it was right thought so but NOT

Third answer at five minutes …….and Now I got it !!

Nice one ! Richard as always, Thank you !

61. I guess it depends if you get your payrise immediately or in lieu. I think there’s more than meets the eye with this week’s puzzle.

I think it set me back that I’m not used to the notion of yearly salaries spread out over a year (I’m a student). Wasn’t really sure where to add the raise (though it’s obvious now that I’ve found out).

63. safc4ever says:

Please remember that the answer to the puzzle is irrelevant to Richard Wiseman. What he is looking for is our comments to see how our minds work when faced with a problem which appears obvious (but may or may not be).

I’m sure he will have fun seeing how serious people get over a puzzle meant as a bit of fun!

1. Jeremy says:

Indeed, that is true. Although I think it’s fair to say that the puzzles themselves have an entertainment value which I imagine Richard enjoys providing.

This one (if you ignore all the ‘I don’t understand the wording’ nonsense), is surely about how one’s ‘intuition’ concerning mathematical ideas is very often wrong. That’s something that is intrinsic to the question, and therefore of interest to someone like Richard Wiseman.

64. Sohvan says:

A really great puzzle this time! My intuition guided me wrong at the first glance too. When I calculated the right answer by looking at the actual payments, I found out I was wrong. It took me a moment to figure out just why the right answer is right after that.

Also interest rates, lengths of employment, and other similar details don’t matter at all. One answer always provides more money as long as the length of employment is at least 1 year.

65. Jarl says:

I still don’t get it. Even after doing the scheduling stuff and all that, I have no idea how the option your intuition would tell you is correct is in fact not the best option. I fell like I am missing something important, so I keep going back to this page over and over to gather clues to what I could have failed to notice.

1. Jarl says:

Just to make sure I have understood this correctly, is the first salary for the second option 52,5 or 55?

2. safc4ever says:

The first payment in both cases is \$50 (6-months worth of the basic \$100 annual salary), as it is 6-months salary BEFORE the first payrise. The second payment will be higher only in the \$5 option because the first rise applies for the second six months. The \$20 payrise is only applied to payments AFTER the first year is completed (i.e. both payments in year 1 will be \$50).

This should help you reach the correct answer (without stating which one it is, as that would be a spoiler!)

3. Jarl says:

Thanks, but that still makes me wounder if the salary after six months is 55 och 52.5 in the second option.

4. safc4ever says:

2nd option each 6 months is \$5 more than the previous ie \$50, \$55, \$60 etc

66. Lazy T says:

I read this puzzle aloud to my brother (a banker) who made an immediate response contrary to my more cynical first idea, he pointed out that “it’s not a percentage increase” and remained adamant in the face of my paper and pencil proof. He needed to stick it through a spread sheet and produce a graph before he went quiet. Then he asked about the bonus scheme!
ta for another great puzzle.

67. Jerry says:

I made an intuitive choice after 10 seconds, then checked it in a spreadsheet. After one year one of the plans emerges as the clear choice, then keeps pulling away, year-upon-year, after that.

1. Jerry says:

That is, the cumulative annual amount pulls away.

68. Joe says:

The answer was very simple to work out in a few seconds, but then reading the question again made me think. The wording of the first paragraph in particular is odd.

69. Marion says:

Well as most weeks I have an answer, am not sure if it’s the right answer but I gave it a go. Will have to check on Monday and will hopefully not be disappointed!

70. rodney says:

less than a minute after a little pen and paper

71. I think this is a social study of some sort.

Since the one answer is so bleedingly obvious, I think Richard is testing us to see how many people can bend this simple “riddle” into answering the wrong thing, just because they think it is a puzzle, when in reality, it’s a 2nd grade math problem with a solution that takes about 3 milliseconds to calculate.

The real experiment is all in the bitching and moaning.

/Z

1. B says:

Thats what I thought. We are part of a psycological experiment. So Richard can write books and earn money.

72. MasterZap got it right – the real riddle is how much people have read into it to make it harder. It’s a real k.i.s.s.

30 seconds of thought – double checked on calc.

73. I like getting the letters, but please will not go telling folks they are likely to obtain about just one a week. I have been lucky to acquire a single a month. You need to do a little better than that….or stop the pinocchio stories at the really least. If one a month is what it is, then that is what it is.

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