coverOn Friday I set this puzzle….

Let’s play a little game.  Please snap your fingers.  Then, 1 minute later please snap your fingers again.  And so it goes on.  Each time you double the time between the snaps, so you snap your fingers again after 2 minutes (from the beginning).  Then snap your fingers after 4 minutes (from the beginning).
Then snap your fingers after 8 minutes (from the beginning).
How many snaps will you make in a year?
If you have not tried to solve it, have a go now.  For everyone else the answer is after the break.
The answer is a surprisingly small 21 snaps!  Did you solve it?  If so, how?
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. 2 to the 19th = 524.288 (i.e. 19 snap in 524.288 min, starting at min 2)
      Add snap at time zero and snap at 1 min ( 2 to the zero power)

    2. Here’s the first way I did it, the dumbhead way:

      0 min – 1st snap
      1 min – 2nd snap
      2 min – 3
      4 min – 4
      8 min – 5
      16 min – 6
      32 min – 7
      64 min – 8
      128 min – 9
      256 min – 10
      512 min – 11
      1024 min – 12
      2048 min – 13
      4096 min – 14
      8192 min – 15
      16384 min – 16
      32768 min – 17
      65536 min – 18
      131072 min – 19
      262144 min – 20
      524288 min – 21 – Very close but still below 525,600 min total in a year.

      Then it occured to me there might be another way to look at it. You must be able to use exponents.

      You have the first snap at the beginning instant of the year
      2 to the 0 power minutes is 1 minute into the year, the 2nd snap
      2 to the 1st is 2 minutes into the year, the 3rd snap
      2 squared is 4
      2 cubed is 8, etc…

      So can you find the largest exponent that puts you at or under 525,600 and then add 2 to it (1 for the first snap and 1 for the second — remember the 2nd snap comes at
      2 to the 0 power which is 1 minute from the start).

      2 to the 10th is 1,024, so that’s way too small.
      2 to the 15th is 32,768, still too small
      2 to the 20th is 1,048,576, wait, that’s way too much, you are over a year

      2 to the 19th is 524,288 which puts us under the no. of mins in a yr.

      So the max exponent that doesn’t go over a year is 19. But you need to add the 1st snap at the start and the 2nd snap represented by 2 to the 0 power.

      So I get 21 snaps. Will be interested to see if I got this right. There is probably a better way than trial and error using exponents.

  1. I did it with a spreadsheet. Put ‘1’ in cell A1 and ‘SUM=A1*2’ in A2 and then drag the formula down the A column. Meanwhile I worked out how many minutes there are in a year and put the two bits of information together. 21. A very pleasing result… No doubt there’ll be some debate because of the wording, but I thought it was clear enough and quite frankly I’m no genius.

  2. The easiest solution is to go BACKWARDS…. and you don’t need a computer for that.
    Realize that the final snap would happen at the end of the year, in which case the one before last would happen 6 months earlier, and the one before that 3 months earlier etc. Using simple arithmetic I came up with 20 snaps…..

  3. What about the contradiction in the question? Who’s gonna kick off a dialogue about that? The question was self-contradictory. Depending on how you interpret the question the answer is either 20 or 21.

    1. I think that everything to say on the contradiction matter was said several times over on last Friday’s thread.

  4. Spreadsheet for the multiples. Pen and paper for the number of minutes in a year… More or less the same as Anne Elk above.

    1. Yes, I figured that out with the help of a calculator. Since I was at university at that time, I didn’t have much time. I’m proud 😉

  5. 1. Calculate the number of minutes in a year.
    (365 days) x (24 hours/day) x (60 min/hour) = 525,600 minutes

    2. You keep doubling the time interval that you snap your fingers,
    so it is an exponential function.

    2^0 = 1 minute
    2^1 = 2 minutes
    2^2 = 2 x 2 = 4 minutes
    2^3 = 2 x 2 x 2 = 8 minutes
    2^4 = 2 x 2 x 2 x 2 = 16 minutes
    2^5 = 2 x 2 x 2 x 2 x 2 = 32 minutes
    2^10 = 2^5 x 2^5 = 1024 minutes
    2^20 = 2^10 x 2^10 = 1,048,576 minutes
    This exceeds the number of minutes in a year (525,600 minutes).
    So, we back up.
    2^19 = 2^20 / 2 = 524,288 minutes
    524,288 minutes fits within a year.

    So, the number of time intervals is 20 (2^0 to 2^19).
    The number of times you snapped your fingers is 20 + 1 = 21.

  6. I got 20, but then realised (having seen the answer) that the first snap was at Time 0, not Time 1. Calculated via a spreadsheet, line 1 = 1, Line 2 = 2, line 3 = 4 etc until I
    got to the number of seconds in a year, 524,288, and the number of lines = number of times, allowing for 1st snap being Time 0.

  7. I tried to do it in my head – roughly.
    6 snaps to get to hours. (1+2+4+8+16+32 = 64-1 = 60 roughly)
    5 snaps to get to days. (1+2+4+8+16 = 32-1 = 24 roughly)
    3 snaps to get to weeks.
    6 snaps to get to get to years.
    That’s only 20 cos I forgot the one at the beginning (classic fence post error!)

    1. Had a similar approach, but i guessed that a year has roughly 512 days so i ended up 21, which was good enough for a rough guess…
      But i had to use my fingers..

  8. After the first 1 minute the rest of the time is a Geometric Progression.
    1 minute, 2 minutes, 4 minutes etc
    So using the formula for the sum of a Geometirc Series a(1-r^n)/(1-r)
    a=1, r=2 then
    1 + (1+ 2^n)/(1-2) = 525600
    then solve for n
    n = log(525600)/log2
    But the value n is only for the Geometric Progression part.
    The total number of ‘Snaps’ is the value of n plus 2 to account for ‘Snap’ at beginning and at the end of the first 1 minute.

    1. General formula: if M minutes have elapsed, then number of clicks so far is
      (2 + log(M) / log(2)) [round the answer down to a whole number],
      eg M = 9, (2 + log(9) / log(2)) = 5.16…, so after 9 minutes, there have been 5 clicks.

  9. But, but it says “double the time between the snaps”, not double the time since the beginning. Richard’s example doubles the time since the beginning, which is a different thing from his instructions.

    Doubling the time between snaps (so they are at 00:00, 00:01, 00:03, 00:07…) you only get 20 in a year. Doubling the time since the start (00:00, 00:01, 00:02, 00:04, 00:08…) you manage 21.

    I claim 20 as the right answer!

    1. I agree about the inconsistency, but given Richard says THREE times “(from the beginning)”, most of us gave that more weight, ie clicks occur at time 0, 1, 2, 4, 8, 16,… and so click N occurs at time (2 ^ (N-1)) minutes. Then 21 is the right answer.

    2. Um, he says 1 then 2 then 4 then 8. I don’t see 3 or 7 mentioned. It couldn’t be more clear

    3. Anonymous – actually it could be clearer, otherwise there wouldn’t be the debate: Ed’s pointing out that the phrase “double the time between clicks” would require, the third click to occur after 3 minutes, but of course that conflicts with other information. Richard probably should have said, “after the 3rd click, double the time between clicks” – or left that phrase out altogether.

    4. Yes, it could have been worded a bit better, granted, but when the words ‘from the beginning’ are used more than once, that should have been a pretty good clue as to what was actually meant; not to mention the inclusion of references to ‘2 minutes’ and ‘4 minutes’… It’s not as ambiguous as some people here seem to want it to be.

    5. Ditto. This is exactly the point I made on Friday. Therefore I can only conclude that Richard’s answer of 21 is incorrect.

    6. Zerbert: It’s 1.) Richard’s answer and 2.) the answer of the vast majority of people who answered the question and posted an answer. You don’t think there might be some room to manouevre, perhaps…?

    7. Given the merest hint of some potential supporting evidence, it’s amazing how many people take delight in arguing that black is white.

    8. The statements in Richard puzzle are contradictory plain and simple. Anyone who doesn’t see that needs to read more carefully. Yes it can be deduced what was actually meant but please don’t put down those that see the contradiction and want to discuss it. Things are always more complicated than they might appear. Many things can be expanded and extended and that makes them all the more interesting. Those that aren’t interested should just not comment and leave those that are to discuss it.

    9. @Steve – Part of the issue might be that those who are determined to prove that white is black are not ‘discussing’ the puzzle at all. The tone is the equivalent of ‘aw, no fair…’ or ‘cheat!’ – when in fact the directions included a simple slip of written language that my twelve year-old nephew managed to wade through without much difficulty.

    10. Anonymussel – Your point is as clear as mud. What on earth are you trying to say. Referencing some 12 year old child is meant to make others feel inadequate in some way ? Is “aw, no fair…’ or ‘cheat” some common saying that I’m supposed to understand ? Maybe your 12 year old nephew told you about it.

    11. @Steve. You referred earlier on to a ‘discussion’ that was being held by those who got the answer to the puzzle wrong. I was pointing out that there was no ‘discussion’ at all: simply a series of complaints that the wording was not clear enough, when in fact it was clear enough if it was given a moment’s thought. I’m sorry that I confused you with my ‘clear as mud’ comment. I look forward to your next splenetic reply.

    12. Hello Anonymussel. Thank you for your reply. I think you have misunderstood the discussion as ‘complaints’. If someone says it clear and another says it’s not clear or contradictory then surely that is a discussion. Not clear enough is an incorrect statement in this case. It is contradictory. If I broke it down mathematical logic it would clearly show a contradiction. Two things that are contradictory cannot be both true. So what’s left for someone to do it to interpret what was meant and that interpretation could be wrong depending on who’s doing the interpretation.

      You say “in fact it was clear enough if it was given a moment’s thought”. Firstly what you say is not a fact and secondly, who’s moment’s thought are we talking about ? Yours obviously ! Other peoples moments of thought maybe somewhat longer or they may not come the same interpretation as you or I. People come from different backgrounds and to simple say ‘if I can get then if you don’t then you are somehow stupid’ which is the impression you are giving.

      Some might call it an argument but I prefer the word discussion as I think that is what you and I are doing now. I didn’t get the answer wrong. I worked out what was probably meant and then got an answer (from making assumptions of what was meant).

      Unlike you it seems, I don’t like statements or questions which allow for interpretation. I have a mathematical background and precision is important to me and many others too. It is because of that, others and my self don’t like contradictions but others like yourself seem to like or happily manage when things aren’t defined sufficiently to disallow ambiguity or interpretation.

      If science worked in the way you seem to like I certainly wouldn’t like to live in the world it produced. The human condition (forged in an environment different to ours today) is well suited to ambiguity or interpretation but the problem with that is we no longer live as we did and it is not the best way of thinking in our modern society. It seems to me that you just don’t see that this ambiguity exists and that interpretation naturally must follow from it. Ambiguity and interpretation leads to misunderstandings and conflict and we all know how that pans out in our world today. If we could just minimize or eliminate ambiguity the world would be a better place.

      I hope this reply is not splenetic. Thanks, I have a new word in my vocabulary now.

    13. @Steve. Thank you for your latest contribution, which (ambitiously) mixes a sloppy attempt at polemic, a hamfisted attempt at psychoanalysis, and even a stab at sociology: all within a blog about puzzles. Quite an achievement… Now that I’ve managed to tiptoe through your typos, and especially now that I have been so thoroughly psychoanalysed by you, I feel humbled to share the same Internet. In fact, we don’t deserve you. A different puzzle blog deserves you and so goodbye. (Oh, and as you’re leaving, a tip if I may be so bold: If you feel compelled to sing the praises of precision in puzzle writing, you might want to think about a proofreader for your own posts.)

    14. Anonymussel – that just tells me that a lot of people didn’t read the question thoroughly. It’s more common that we like to think. Even Richard is human, and we all make mistakes sometimes.

    15. Anonymussel steve’s point about contradiction and interpretation is spot on. In general conversation, yes of course we can map out the meaning of what people say when they slip up or use an incorrect word. You can’t do that in math, and you certainly can’t do that in a math oriented puzzle blog. You say your 12 year old nephew was able to navigate his way through, congratulations, I am glad he got the same answer as Richard, but based on the wording given, he was just as likely to choose the other possible answer. And he ends up just as wrong as Richard is. We’re only human after all. And it’s not the first time Richard has been wrong, I am sure.

    16. @Anonymussel. I find it hard respond to your latest reply. I’m not sure where to start. I just can’t make sense of all of it. You are either deliberately being mischievous or honestly believe what you say. I can only think that you just don’t have a science background.

      I think I would agree with your analysis of my attempt at explaining myself. It probably was ambitious, sloppy and probably an overall poor attempt but I’m learning and I thank you, in some sense, for helping me (this is honesty by the way and not sarcasm). I would honestly be grateful if you could let me know what my typos were as I’d like to be aware of them. Yes, honestly I really would, as you’ve already proof-read (?) it.

      I am really sorry if any of my comments were seen by you as some sort of attack on you. If there were then I made an error of judgement in that area.

  10. I wanted to find a pen and paper solution that avoided multiple iterations (Excel etc.) by trying to deduce the order of magnitude (base 2) of the last interval. The last number of a doubling progression is always one greater than the sum of all preceding numbers so I figured any interval of half a year or less would still fall within the year. And the last interval NOT to exceed six months would have to end within the second half the year (because it would have to be greater than three months (or else you could double it again so it wouldn’t be the last not to exceed six months) and so with the sum of all preceding intervals greater than six) so the next interval would have to take you out of the year. So working out the number of minutes in six months and finding the highest power of two that comes under this number should lead to the answer.

    That said I failed to add in the extras and got it wrong!

    1. Amazing how some you complicate such simple information. How do get on with the other puzzles?

  11. I worked out how many minutes in a year (365 x 24 x 60) then counted how many times I needed to halve, using a calculator, to get down to 1 or almost 1, (20), then added 1 for the very first click.

  12. Next week Train A leaves London at 9 AM…

    This isn’t a puzzle. It’s a straightforward math problem. Let’s get back to something that makes you think a bit.

    1. The reason it’s a “puzzle” is because the answer is surprisingly small. Intuition might indicate that the answer would be hundreds or thousands. Having worked out the answer and comparing it to expectation does make one “think a bit.”

    2. I guess perhaps it was a surprise to some. Anyone that is familiar with old chess board doubling problem knows instinctively that the answer would be small.

  13. Since the doubling is nicely represented in binary, I used the Calculator app on my Mac in Programmer mode. 365 * 24 * 60 = 525600, and it handily displays the binary equivalent as 1000 0000 0101 0010 0000. That’s 20 digits long, equal to the 20 snaps that will take place over the year. Add in the starting snap at the beginning to get 21. Add in an extra “error bar” snap if you want to cover the puzzle being poorly worded.

  14. Richard has given the wrong answer to his own puzzle. The answer is not “a surprisingly small 21 snaps!”; the answer is 21. It would have been a surprisingly small number if we had expected it to be something like 400, or 8537, or 29 billion and 8, but we weren’t.

  15. (a) nobody has yet mentioned the coincidence that the length of the year is very close to the 21st snap, so that the final snap actually happens on 31st December (at 2:8am, to be precise) and not at some dull time in September or November or whatever.

    (b) I am surprised that a lot of people in the thread say that they calculated the number of minutes in a year as being 365 * 24 * 60. I calculated it by the much more simple and obvious method of dividing 31,536,000 by 60.

    1. i think it’s far more interesting that the length of time from the beginning to the last snap is so close to the length of a year, that you actually need to calculate the number of minutes in a year to be sure, and that the answer turns out not to depend on what kind of year you use (calendar or astronomical), even if it’s a leap year.

      not to be pedantic, however, the answer does not hold if you use a lunar year, which gives 20 snaps. while lunar years are atypical, the problem does not rule them out as phrased.

      and not to be more pedantic, but (a) nobody says “31st December” (it’s “December 31” or “December 31st”), (b) there is no such time as 2:8am, and (c) there is nothing simple and obvious about dividing the meaningless number 31,536,000 by 60. perhaps you meant to show off that you think you know the length of the year in seconds, which on average is more like 31,556,925.

    2. As fascinating as ever ctj, but we are still waiting to hear of your own spiffing solution to the 21st June Friday Puzzle.
      You may recall that you solved this brain teaser in about five minutes whilst dissing the customary solution normally found on Google…’s still not too late to share your answer with us.

    3. the coins that form the V never touch the bottom coin. they come close, but if you assemble it you will see that they actually don’t touch.

      the solution: focus on the two coins (A and B) that are atop the bottom coin. take the fourth coin and stand it on its edge atop the bottom coin so that it is tangent to coin A (making an L shape), and its edge touches the edge of coin B. do the same thing with the fifth coin, making an L with coin B and touching the edge of coin A. the fourth and fifth coins will be roughly parallel but not directly facing each other. now tilt the fourth and fifth coins together until they touch, so that the L becomes more like < (the angle looks to be roughly 85 degrees).

      viewed directly from above, the arrangement looks like a an infinity sign on top of a figure eight and tilted by roughly 60 degrees.

    4. In answer to ctj:

      (a) The vast majority of ordinary normal people say “31st of December” which is written as “31st December”, and only a tiny minority of deranged people say “December 31st”.

      (b) I wrote “2:08am”, not “2:8am”. My comment has obviously been hacked, infiltrated and edited by a malevolent hippopotamus from the Andromeda Galaxy.

    5. To ctj: The use of ’31st December’ is very common in the UK in written English. Spoken out loud it’s usually rendered ’31st of December’.

    6. With reference to “nobody says “31st December”, in the UK we do say that (but with an ‘of’ in between) just as we (the British … there may be some variations) write 31/12/2013 rather than 12/31/2013.

  16. Place the grain of wheat on the first square in the chess board, and TWO in the second quarter, and four in the fourth box and …..In the end how much grain in the fourth quarter

  17. i am confused. does this 21 ans take into account the extra gap that occurs (and is thus doubled in each time after) when clocks go back in march?

    1. Dharma, the clocks SPRING forward in spring, and FALL back in autumn. The net effect is that it makes no difference over a year.

    2. yet one hour difference in the SPRING (forward or backward, make no real difference here) will equate to 100,000 hours in the AUTUMN once ten doublings take place.

      that will affect the Case, yes?

  18. Most easiest way:
    Assuming in a normal year of 365 days of 24 hrs of 60 mins (not that it matters here), we have 365x24x60 = 512 640 mins. Then you start divide by 2. Count the number of division make till you hit less than one mins (the first interval). Add one to that number (the starting snap). And you get 21.
    Why this work, number theory. I would spare you the detail. But the following generalisation will always work:
    Let the startin interval be x mins, the number of snap be n and the ratio between each snap be y mins, we then can generalise the following formular:
    (Time limit)/y^n < x
    Now since time limit, n, y and x are all counting numbers, we can re arrange to get this:
    (Time limits)/x < y^n
    Then take the log to the base y:
    log base y [(time taken)/x] < n
    Then round to the nearest number.
    Quod Erat Demonstrasdum
    By a year 10 student.

    1. Was the minutes in a year calculation also done by a year 10 student ? I see no point to making references that some child can do it therefore if you can’t then you are somehow stupid. Please keep children out of this, unless you are that year 10 student.

      Also “Quod Erat Demonstrasdum” is normally written QED. Are you trying to make the none Latin scholars feel inferior ?

  19. It’s simple. 365x24x60=525600
    Then either Log of 525600 to the base of 2=19. Add the click at t=0 and the click at t=1min as 1=2^0 to get 21.
    Or log(525600)/log(2).

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