On Friday I posted this puzzle….

Given a sack of sugar, an unbalanced scale, and two 5-pound weights, measure exactly 10 pounds of sugar.

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

Put both of the weights in one pan and pour the sugar into the other pan until the scales balance. Then remove the weights and pour sugar into the first pan until it balances again. The first pan will contain exactly 10 pounds of sugar.

Any other 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.

43 comments

  1. Almost the same as my solution:
    – Put sugar on one side to make it balanced
    – Put the weights on the same side as the sugar
    – Put sugar on the other side to make it balanced: 10 pounds.

    1. Sorry, I’ve been away for a few days attending the 2012 Pedants Conference. Have there been any messages been left for me?

  2. The real puzzle: Why was a straightforward procedure of balancing and weighing considered a puzzle in the first place?

    (No need to put the weights in one pan before you balance the scale, btw.)

    1. The imbalance may have pegged the position at the max, so that may not be accurate. If the imbalance is over 10 pounds, the position wouldn’t even change when you added the weights!

    2. I think it has to be taken as implicit that that the imbalance is less than 10 pounds. if it was greater, Richard’s own answer wouldn’t work if he put the weights on the lighter side to begin with.

      My alternative answer (which also presumes the imbalance X is within +/- 10 pounds) is to weigh suger against a 5kg weight placed on one side of the scales first, giving you 5+X kg. Then weigh sugar against a 5kg weight placed on the other side, giving you 5-X kg. Then add the two lots of sugar together. This wastes less sugar overall than Richard’s answer.

    3. This was my solution also.. It doesn’t matter the position of the pans, just as long as they are brought to same points after the addition of the weights and sugar.

  3. If the unbalance is less than 5kg:
    – Put a weight on one side, and add sugar to the other side until the scales balance.
    – Take that sugar off, and into your final container.
    – Move the weight to the other side, and add sugar to the opposite side until the scales balance.
    – Take that sugar off and into the final container.
    – You now have 10kg of sugar exactly, with no wasted sugar…

    If the unbalance is between 5kg and 10kg, then you first add a 5kg weight to the lighter side and leave it there. You then repeat the steps given for the first case. Still no wasted sugar.

    If the unbalance is between 10kg and 15kg then do the same, but with an extra 5kg added.

    If the unbalance is greater than 15kg, then you should do similar to Richard’s answer…

    1. Richard’s method uses a lot of sugar. A simple method that involves less use of sugar, for imbalances of less than 10kg, is as follows.

      For a small imbalance (<10kg), add sugar to the light side until it balances. Then add the weights to the same side and put sugar on the other side until it balances.

      If the imbalance is more than 10kg, and want to keep sugar use down, it is best if you can make two separate piles in the same pan. Add both weights to the light side, plus additional sugar to make it balance. Then take the weights off and make a separate pile of sugar on the light side until it balances.

      It is far from clear that an imbalanced scales, once put into balance by addition of a weight to one side, will remain in balance if an equal weight is then added to each side. It depends what is wrong with them. This is because scales react not only to the weight put on one side, but also by its distance from the pivot. If the pans were no longer symmetrical about the pivot, all this would be a waste of time.

  4. I did it differently. I put sugar on the higher side until it was at 0, Then added the weights to the same side and sugar to the empty side to measure it out.

    At first I was going to put the weights on the low side, sugar on the high side until it got to 0, then remove the weights and spoon out the sugar until it got to 0. The spooned off sugar would be the right weight.

    Both work, my second thought was a more elegant solution, but not as elegant as Richard’s solution

  5. Yeah this answer does seem over complicated really. Balance the scale by adding sugar to one side until it’s level. At that point you no longer have an unbalanced scale, you have a normal scale. So you can then stick weights on that side and sugar on the other in the normal way to measure out your 10lbs.

  6. I think next week’s puzzle should be ‘Why do some people visit this site every week, only to moan every week about how much they hate the Friday puzzles?’

    1. We visit because in the old days, long before you were born, you young whipper snapper, there were some good questions. .
      Words can barely describe the level of intellectual challenge provided by Richard’s Friday puzzle. Young men (and women) could barely sleep at night on a Thursday in expectation of what was to come…….that fought with us on Crispin’s day etc.
      We harken back to that bygone age when men were men, women were women and LBGT were LBGT etc.
      You young uns dont know you are born with your ipads and your twitters…………

  7. Clever, clever…!
    But theres many waste of sugar ! I have a solution with no waste of sugar:
    1. At the start, mark the position of the pointer
    2. Do the weighting as usual, but just use the new made mark instead the zero mark.

  8. Put the weights on opposite sides then balance the higher side with sugar.

    Move both weights to the sugared side and pour sugar into the other until balanced again.

    The side without weights now has the required amount of sugar.

  9. tAKE THE FOX AND THE GOOSE TOGETHER; RETURN WITH THE SUGAR; THEN TAKE THE WEIGHTS ADN THE OTHER GOOSE ……………………..

  10. Duh. Step 1: Use some sugar to get the scale to balance. After that you are free to measure as you see fit. Not much of a puzzle this week.

  11. All the answers that start by adding a little sugar in order to first balance the scale are wrong. They only work if the arms of the scale are equal in length, and the original imbalance is caused by the pans being of different weights. If the arms are of different length, this method fails whereas Richard’s method still works.

    1. I’m sorry but I don’t see how Richard’s answer differs from the ‘adding a little sugar… first…’ method. Surely the difference between 10lb and all the sugar Richard adds to the other pan is equivalent to the ‘little sugar in order to first balance the scale’.

  12. To Eddie: For the sake of argument let say that one arm is twice the length of the other (and for this thought experiment the arms are weightless), and the pans each weigh 1lb. Then you have to add one pound of sugar to the side with the shorter arm in order to initially balance the scale. After that initial calibration, the scale will balance whenever there is twice the weight on the side with the shorter arm as on the side with the longer arm. So if you after the initial balancing adds 10 pounds on one side, you will have to add either 20 pounds or 5 pound to the other side for the scale to balance again (depends on which side you put the 10 pounds).

    Richards answer works because his solution is to put the 10 pound weights on one side, then add sugar on the other side in order to balance the scale. Then after that he removes the 10 pound weights and add sugar on that side in order to rebalance the scale. That way the 10 pound weights and the sugar to measure up is on the same side of the scale.

    Using this we can be sure that the measurde suger weighs as much as the weights. But we don’t know how much weight is on the other side of the scale.

    1. What does Richard mean by ‘an unbalanced scale’? Does he mean that the pivot is not in the middle and the restoring moments required to balance the scale are not equal. I’ve never heard of a scale which requires unequal weights to balance it.

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