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10 Of the Best…?

Posted by Karl Sharman on January 18, 2011 – 10:42 am

1. If 3 salesman can sell three stoves in 7 minutes, how many stoves can six salesmen sell in seventy minutes?

2. Which number when added to 5/4 gives the same result as when it is multiplied 5/4

3. Using only 3 cuts, how do you divide your round birthday cake into 8 equal slices?

4. A fish is fifteen inches long. Its head is as long as its tail. If the head were twice as long as it really is, the head and tail would together be as long as what’s in between. How long is each part of the fish?

5. What would be the next number in the following sequence -
11 1,331 161,051 19,487,171 ???

6. The square of 13 is 169. Take the last digit of the square, 9, and place it in the middle, making 196. This is the square of 14, the next number above 13. What are the next numbers which also have this property?

7. There are several ways to come up with 100 by using all the digits 0 through 9. One way is: 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + (8 x 9) = 100. How many more ways are there using plus, minus, divide, multiply and of course brackets….?

8. If 1/2 of 16 were 13, what would 1/3 of 32 be?

9. You have a huge box of beautiful decorated tiles, enough to provide a border in two rooms. You really can’t figure out how to arrange them, however. If you set a border of two tiles all around, there’s one left over; if you set three tiles all around, or four, or five, or six, there’s still one tile left over. Finally; you try a block of seven tiles for each wall, and you come out even. What is the smallest number of tiles you could have to get this result?

10. Of all these questions, which one was impossible?

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  1. 1. DP Said:

    1) 60
    2) 5
    3) I don’t know
    4) 3-9-3? I’m assuming you don’t mean twice the head plus the tail is what is in the middle then, but with regular lengths.
    5) I don’t know
    6) Not possible
    7) I don’t know
    8) 12?
    9) I need to do a little more math…
    10) #6 is not possible

  2. 2. DP Said:

    #9 is 301

  3. 3. DP Said:

    oh i know #3 now. you can cut it into 4 wedge shapes (2 cuts), stack them all and cut into smaller wedges (3rd cut)…
    you can cut it in half so that you have 2 shorter round cakes (1 cut). Then leaving them stacked, cut into 4 wedge shapes (2nd and 3rd cuts)

  4. 4. NIK Said:


  5. 5. Krazeedude Said:

    1.) 3*2*10 = 60

    2.) X1.25 = 1.25+x
    1.25(1.25x) + 1.25X2 =0
    1.25X2 = -1.5625

    3.) Cut into 1/4s, stack and 3rd cut is through centre

    4.) 1:x:1
    2:x:1 (x=2h+t) x=2+1
    1:3:1=5 15/5=3 1 part=3

    5.) 11^1 = 11
    11^3 = 1331
    11^5 = 161051
    … Fail commences

    6.) God knows
    7.) CBA
    8.) 16/2 = 13 (only last digit is halved)
    32/3 = 30+2/3
    9.) Not a clue
    10.) 6 feels pretty dodgy but for mathematical reasoning, 8 cannot exist or happen and is therefore impossible.

  6. 6. Krazeedude Said:

    NIK; I took the 3 original stoves, multiplied by to, since the number of salesmen and been multiplied by 2, and then multiplied by 10 since the time given has been multiplied by 10.
    3*2*10 = 60

    What calculation are you doing?

  7. 7. DP Said:

    Thanks Krazeedude. I solved #1 the same way.
    Also, I forgot that this site likes to make smiles, so me 8) was supposed to be #8.
    I’m still unsure what the numbers and spaces indicate on #5, and have no clue the logic on #8.
    @ Krazeedude: I think you complicated #2 a little too much.
    1.25x = 1.25 + x => 0.25x = 1.25 => x = 5

  8. 8. NIK Said:


  9. 9. Jappa Said:

    just saying, but nik, is caps lock necessary?
    cuz it seems somewhat rude, and its not like everyone is arguing against DP or chris or anyone, like in girls, girls, girls, so you don’t need to ‘yell’ for attention?

  10. 10. Chris Said:

    5. 23,579,47,691 which is what Krazeedude’s calculator coudn’t handle. The spaces separate the numbers, the commas are the groups of three digit thousand, millions …

  11. 11. Chris Said:

    1. One salesman can solve one stove in 7 minutes. So that’s 1/7 th stove/salesman/minute. So 6 * 70 * 1/7 = 60.

  12. 12. Chris Said:

    9. Let n be the number of tiles.
    We have n = 1 (mod 2,3,4,5,6) = 1 (mod lcm(2,3,4,5,6)) =>
    n = 1 (mod 60)
    So n = 60k + 1 = 0 (mod 7)
    60k = 6 (mod 7), added 6 to both sides
    4k = 6 (mod 7), as 60 = 4 (mod 7)
    8k = 12 (mod 7), doubled both sides
    k = 5 (mod 7)
    so k = 5
    so n = 301

  13. 13. Chris Said:

    6. For 3 digit numbers abc = x² we need acb = (x+1)²
    So 10(c-b)+(b-c) = 9(c-b) = 2x + 1
    For c-b = 1,3,5,7,9 => x = 4,13,22,31,40.
    But there is no integer x for c-b = 0,2,4,8
    4 is too small and 40 too big for a 3 digit result
    Trial and error shows that only x = 13 works.

    As must move to the middle, we need only consider numbers with an odd number of digits. Consider an arbitrary odd string of digits M:a:N:b where length M = length N + 1
    Note b must be greater than a.

    Let a be in the 10^n position, then (10^n)(b-a) + a-b = 2x+1
    ((10^n)-1)(b-a) = 2x+1
    e.g. 99(b-a) = 2x+1
    if b-a = 1,3,5,7,9 then x = 49, 148,…
    This doesn’t work for 49 and 148 cause overflow
    e.g. 999(b-a) = 2x+1, b-a =1 => x = 499 and overflow
    clearly that never improves.

    13 is the only number that can do the trick and this problem is insoluble.

    So that means that either 8 or 10 is insoluble.

    I can’t fathom 8 (I’ve tried some variations on mixed up number bases). But I know that Karl plays fast and loose with mathematical notation, so Krazeedude may be right and 10 is insoluble. Or …

  14. 14. Chris Said:

    Aaargh I mucked up that last one. With 99(b-a) we need 316 > x ≥ 100
    b-a = 1,3,5,7,9 => x = (49 too small),148,247,(346 – too big)
    And neither 148 nor 247 work.
    skip 999(-a) as wouln’t be an odd length string of digits
    9999(b-a) = 2x+1. 31622 > x ≥ 10000
    b-a = 1,3,5,7,9 => x = (4999 too small),14998,24997,(34996 too big)

    14998² = 224940004, 14999² = 224970001
    24997² = 624850009, 24998² = 624900004

    999999(b-a) = 2x+1. 316228 > x ≥ 1000000
    b-a = 1,3,5,7,9 => .., 1499998, 2499997, …

    e.g. of what’s happening
    1499998² = 2249994000004, 1499999² = 2249997000001
    2499997² = 6249985000009, 2499998² = 6249990000004

    and these patterns more seem to go on for ever.

    Not surprising. We will only ever (and always) be able to choose b-a = 3 and 5. For (b-a) = 3, we get 1499….998. So the square will end in 4. But 1499…998) = 1500….000 – 2, so the square will be
    225000….000 – 60000…. +4 = 224999…9994000…0004

    It’s easy to make similar observations about the 24999…997 mubers and the x+1 versions.

    So, phew!, Karl has shown the only possible example.

  15. 15. Karl Sharman Said:

    Chris…. maybe not, there may or may not be other examples…
    And “fast and loose” with mathematical notation? I am mortified!!! ;-)

  16. 16. Chris Said:

    Hi Karl. I was thinking of your “Impossible Number” post. You allowed base 10 string catenation as a mathematical operator ;)

  17. 17. Karl Sharman Said:

    Chris, originally, I hadn’t considered putting the digits together to make a number in it’s own right – string concatenation – until SP pointed it out and I felt that the question did only comment on using the digits… oh alright!!! It might have been a bit liberal….
    Aside from that, the only other example I have is somewhere between 100 and 200… if that helps any! I did it on an excel spreadsheet – so no real maths there!

  18. 18. Euclid's Brother Said:

    #3: First cut in half, then make a perpendicular cut, so you have 4 equal pieces, then make a horizontal cut, seperating the top from the bottom equally.

  19. 19. Chris Said:

    Hi Karl. I knew that it wasn’t yout original intention. :)

    But I still can’t fathom question 8.

  20. 20. Dual Aspect Said:

    For No 8. I get 8.666667

    If 13 is half of 16 then 26 would be half of 32. (This is dubious depending on the questions definition of “1/2 of” but I’m guessing that this is where the question is leading us).

    1/2 x 2/3 = 1/3

    so… 26 x 2/3 would give 1/3 of 32 = 8.666677

    BTW Karl – could you throw some light on the answer of your “Adds up to a thousand” post. There are 3 different answers for the number of triplet combinations and I’m intrigued to know which is correct.


  21. 21. Dual Aspect Said:

    Sorry, the above is complete rubbish of course.

    Same reasoning, but 26 x 2/3 = 17.3333333

    So that is my answer


  22. 22. Chris Said:

    10. Assuming an integer result, then Q8 is either very silly or impossible.

    Assuming an integer result and playing with number base representations, let the given number be in base A and the result be in base B.

    1/2 * 16 = 13 => 1/2 (A+6) = (B+3) => A/2 +3 = B+3, so A = 2B
    Then 2/3 (3A+2) = 2A + 4/3 and that cannot be be represented in any integral number base system.

  23. 23. Chris Said:

    Karl, I think you are practising trickery. I’ll recheck 6 in view of your comment. I’ve only just twigged that if 6 is unanswerable and 8 is unanswerable then 10 is not a sensible question (on purpose) – and other stuff along those lines.

  24. 24. Krazeedude Said:

    My opinions on everything…

    3. Euclid’s brother’s answer to #3, in my opinion, is the best, it solves the problem without the need to get too involved.

    5. I am so annoyed that I misread question 5, I got that it was 11^n where n is increasing in odd numbers, I just got confused at the comma’s, which is a bit crap of me really.

    Jappa: all of NIK’s posts, including his name, are in caps lock, some people just prefer it, not out of attention grabbing or anger, simple aesthetic pleasure.

    Chris: Your theory on question 8 is more likely. Mine is very simplistic, yours sounds like it was written by someone who thinks. One of us could be correct with #8, leaving #6 as the unanswerable, which isn’t surprising, and thus making #10 answerable as well.

  25. 25. Chris Said:

    9. Is impossible. If you only had enough tiles to border the rooms two tiles deep with one left over, you don’t have enough tiles to do 3,4,5,6 or 7 tiles deep.

    What does “each corner” mean?

  26. 26. Chris Said:

    I find upper case is painful to read (seriously).
    It is bad netiquette.

  27. 27. Dual Aspect Said:

    The difficulty with No 8 is that the questions redefines the phrase “1/2 of” to mean something different to what we understand it to be but we are only given one example of how the new definition of “1/2 of” works.

    This being the case there are an infinite number of variations of what “1/2 of” could mean according to the question.

    So, logically, to solve the puzzle we must use the information that we are provided with to somehow relate “1/2 of 13″ with “1/3 of 26″ without needing to understand what “1/2 of” is referring to.

    The only way I can see of doing this is by assuming that 1/2 of 26 must equate to 2 x 1/2 of 13, which in our traditional definition of “1/2 of” would be correct but which we cannot categorically prove for the new definition of “1/2 of” which we do not know.

    Given that the question requires us to make at least one assumption to solve the puzzle, this seems (to me) to be the logical solution.

  28. 28. Karl Sharman Said:

    No-one appears to have worked on old original No.7 yet….

    Did anyone dive in and start answering the nice easy questions at the beginning, before reading to the end?

  29. 29. Karl Sharman Said:

    Chris- post 25 – updated question to read each wall. Don’t know what I was thinking at the time, although I still had to read it a few times to work out what was wrong! Typical D’oh moment!

  30. 30. Chris Said:

    Hi Karl. I still don’t understand what question 9 means.

  31. 31. Chris Said:

    Q9. If I assume that there are just blocks which are multiples of 2,3,4,5,6 on each of 8 walls then an even number of tiles will have been used. But there’s always 1 tile left over, so there’s an odd number of tiles altogether. When you use 7 tiles per wall you will have used an even number of tiles (must be divisible by 8). But as there was an odd number of tiles, there must still be at least one tile left over.

    So Q9 is impossible. But I can’t make a sensible interpretation of what is meant by “borders”.

  32. 32. Karl Sharman Said:

    A border would be a row of tiles (1 deep, 2 deep, 3 deep etc) running around the whole room (4 walls). But you’ve already answered the question correctly, post 12, so Q9 is possible.

    301 tiles is the smallest number that will give you a remainder of 1 when divided by 2, 3, 4, 5, and 6, but divided by 7 leaves no remainder.

  33. 33. Nik Said:

    Sorry my bad with the caps, need to use them at work and completely forgot to hit caps lock before replying.

  34. 34. Chris Said:

    Hi Karl. OK we both use “border” with the same meaning. But even if I gnore the complications of two rooms with four walls each, if a two tile border was achieved, then it would be two rows of 150 tiles = 300 tiles. To tile the same length 3 tiles deep would require 3*150 tiles = 450 tiles.

    So Q9 isn’t possible for that reason alone. But it also isn’t possible as there has to be both an odd and an even number of tiles – there is no such number.

    Unless ^^ the tiles weren’t squares or weren’t touching.

    Even then, it’s still not possible.

    I’ve made many assumptions in the following.
    We need n = 1 (mod 2*8,3*8,4*8,5*8,6*8) = 1 (mod 480)
    Then n = 480k + 1 = 0 (mod 56) and that can’t be solved.
    e.g. 480k = 55 (mod 7*8 = 56)
    => 32k = 55 (mod 56)
    times 7 => 0 = 49 (mod 56) and is a contradiction.

    I guess that if the rooms weren’t rectangular and were of different sizes and gaps were allowed between the tiles then just maybe that it could be done, but that has all got to be rather silly, so I’m not going to examine that possibility.

  35. 35. Chris Said:

    Hi again Karl. Post 12 was written without having properly considered the question. I just assumed it was “the usual” problem. Q9 is impossible no matter how you try. 301 doesn’t do the job at all. Methinks that you made the question up :)

  36. 36. Karl Sharman Said:

    Chris, you didn’t think the impossible question would have been an earlier one? I thought post 28 might have hinted at it. The easy answer as you have surmised is 301, but with a little more thought, or 301 tiles, 2 rooms and a lot of time on your hands it just doesn’t work. So after much obfuscation and inveiglement on my part, I shall award you the points for Nos 9 & 10.

    On the plus side not all questions have been answered yet. Nos 6,7 and 8 definitive answers are proving elusive it appears!

  37. 37. Chris Said:

    Hi Karl. Q6. I’ve assumed swapping digits. Do you actually mean to insert the digit? e.g. if original number is abcde is the new number abecd? If so, that could be fiddly. I also assume that it’s only the units digit that’s moved/inserted and that it is n² and (n+1)² but not e.g. (n+2)².

    Q7. It strikes me that there could be quite a few ways of doing it, but you’d never be sure that you hadn’t missed one. But I haven’t tried. If you allow string catenation ;) , then there’s probably loads of ways. I’m going to stick my neck out (without having tried at all) and say you’ve given the only possible way of doing it with the rules you’ve defined.

    Q8. There aren’t enough clues to let us guess what to do. It certainly doesn’t look like you’ve use anything in even a semi-normal way. I can give a totally stupid answer. First add 10 to the second number: 1/2 (10 + 16) = 13. Likewise: 2/3*(10 + 32) = 28. If you have redefined the meaning of words and symbols, there may be an infinite number of possible answers. We need at least one more example to have a hope of doing this one. We also really would like to know if the answer is an integer.

    It’s seems unlikely that anyone is going to get this one without a clue.

  38. 38. Chris Said:

    Q6. The next one is 157² = 24649 with 158² = 24964.
    I note that I have put the end number into the middle.

    I’ll see if I can find general solution.

  39. 39. SP Said:


    half of sixteen = 13 letters (not counting spaces)
    third of thirty two = 16 letters

    My guess is 16.

  40. 40. Karl Sharman Said:

    Chris – Q7. – ((7×9)+(8×4)+5+((1+2+3)x0)=100

    So probably a few more ways…?

  41. 41. Chris Said:

    Q6. I can se no easy way to find a general solution. I usually find that these oddities based on decimal representation have practically intractible solutions.

    Q7. As I feared. There may be many ways. I’m not going to try – life’s too short.

    Q8. Aaaaargh. I’ve be doing 2/3 of 32 instead of 1/3. Previous 28 is now 14. i.e. 1/3(10+32). But I give up on that one. I like SP’s idea, post 39

    Q2. x + 5/4 = x * 5/4 => 5/4 = x *1/4 => x = 5

    Q3. h + m + t = 15, h = t, 2h + t = m
    Subst h = t into first and third equation =>
    2h + m = 15, 3h = m => 5h = 15 => h = t = 3 and m = 3*3 = 9.
    So h:m:y = 3:9:3

  42. 42. DP Said:

    I like SP’s solution in post #39 for question 8.
    “Half of sixteen” is 13 letters (not counting spaces)
    now, is it “one third of thirty two”, because we didn’t use “one half” on the first part. If it is “one third of thirty two”, my answer would be 19. there are 19 letters, no spaces.
    if it is “third of thirty two”, the answer is 16 as SP pointed out.
    Just to be different, i’ll go with 19.

    @ Nik: do you work in CAD a lot? i know it is industry standard to use CAPS on everything. I sometimes forget too, especially since I do this at work as well.

  43. 43. DP Said:

    Karl, for Q7, do you mean to say using every integer 0 through 9, or digit? there will be much fewer answers if you cannot combine digits. If you were able to, you could end up with solutions like 98+3-7+6+1-5+4+2*0. there would be too many to try. I’m with Chris in that i will never attempt to answer this question if that is the case.

  44. 44. Chris Said:

    Hi DP. Because Karl would probably not wish to incur my wrath (LOL), I’m pretty sure that he wasn’t allowing catenation.

  45. 45. Karl Sharman Said:

    Chris is right – no catenation, and all numbers must be used. Although for you, and only you DP, I might just allow catenation. Nope, best not! Chris would not be pleased if I started making exceptions ;-)

  46. 46. Karl Sharman Said:

    1. 60 – DP
    2. 5 – DP
    3. 2 cuts, like an X on top, and then cut horizontally across the middle – DP
    4. The head and tail are each three inches long; the rest is nine – DP
    5. 2,357,947,691. The numbers are 11 to the first power, 11 to the third power, 11 to the fifth power, 11 to the seventh power; so the missing number is 11 to the ninth power – Krazeedude spotted the sequence, Chris had the bigger calculator!
    6. 157 squared is 24649, and 158 squared is 24964 – Chris
    7. Loads
    8. 17 1/3. If you set it up as a proportion it’s easier to see
    9. 301 tiles is the smallest number that will give you a remainder of 1 when divided by 2, 3, 4, 5, and 6, but divided by 7 leaves no remainder but that doesn’t work for the 2 rooms… Chris
    10. 7 can be done – I have no idea of how many possibilities there are – in 5 minutes I came up with 20+.
    No. 9 is the one that looks easy, but as Chris, our resident tiling expert (good rates, freindly service and he prices up his jobs to the exact tile) can attest – is not very possible.

  47. 47. Chris Said:

    Karl. In detail, how do you get 17 1/3?

    (This has gotta be good;))

  48. 48. Dual Aspect Said:

    Chris – see my posts 20 & 21 (and also 27)

    I got 17.333, although I made a typo in the first post.

    Karl – Where’s my name in lights???


  49. 49. Chris Said:

    Nice one Dual. But you’ve assumed that 2/3 plays ball, even though 1/2 doesn’t ;) You definitely should get the credit for that.

  50. 50. Dual Aspect Said:

    Yeah, I agree,

    It’s tenuous, as I acknowledged in my posts, but it seemed the only logical option to make that assumption.

  51. 51. Karl Sharman Said:

    This doesn’t come out right, but here goes…

    1/2 (16) = 1/3 (32)
    13 x
    Cross multiplying gives: 8x = 416/3 => x = 416/24 => x = 17 1/3

  52. 52. Chris Said:

    LOL. Karl, if 1/2(16) = 13 and 1/2(16) = 1/3(32) then I reckon that 1/3(32) = 13.

    I hope that you don’t mind me taking the liberty of saying that you’re taking liberties with the whole of mathematics as I understand it ;)

  53. 53. jhanvi Said:

    #6 is157 squarred is 24649 and 158 squarred is 24964

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