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There is an iron kettle-bell which weighs 6 kilograms, an infinite amount of sugar, infinite amount of packets for it and a weighing scale which has two weighing pans.

The scales are in equilibrium if the ratio of masses on the first to the second cup is 3:4. You are allowed to put any weight you have on the scales. You can also add a packet of measured sugar to one of the cups which will let the scales come to an equilibrium. The packets with weighed sugar may be used in the next weighings.

Is it possible to measure 1 kilogram of sugar?

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3 Answers 3

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Sure you can!

Because...

First, put the 6kg on the '3' side of the scale, and measure out 8kg of sugar. Take the 6kg weight off, and measure out 6kg of sugar. Now put them together and you have 12kg - put this on the '4' side. This will be balanced out by 9kg on the '3' side. Put your 8kg of sugar there, and an empty bag. Slowly pour sugar into this new bag until it balances, and there will be exactly 1kg of sugar in the small bag.

I'm assuming here that...

I can put more than one bag on one side of the scale, which I feel is reasonable given than bags can mould around each other. If not (for example, if the bags are hanging on a hook with only room for one handle), then a solution is to put a big bag on the hook (infinite sugar = infinitely sized bags) and put the two smaller bags inside it.

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  • $\begingroup$ Then add 1kg of self-raising flour, 1kg of butter, 18 eggs, 2T baking powder, 2T of vanilla, and 1/2t of salt. Feeds... about 20 people. $\endgroup$
    – eedrah
    Oct 22, 2017 at 9:23
  • $\begingroup$ @boboquack - sure. Could you point me to a reference on how I should format my answers? You want me to remove all the spoilers, or just add a couple of words at the start as introduction? $\endgroup$
    – eedrah
    Oct 22, 2017 at 9:26
  • $\begingroup$ Oh, the spoilers are fine. But a page of yellow empty blocks isn't very friendly to readers, and some readers might want to just see the answer without the process. My rule of thumb is that all spoilers should have at least one line of text above them that would allow you to work out what the content of the spoiler is, and spoilers should represent one idea. For example, the answer to a question and how you came about that answer are different, but how you came about an answer and the proof to the answer can usually be merged. $\endgroup$
    – boboquack
    Oct 22, 2017 at 9:32
  • $\begingroup$ that's a good mantra - I like it. $\endgroup$
    – eedrah
    Oct 22, 2017 at 9:33
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The answer is:

Yes

Proof:

• Put the kettlebell in the first cup, put into the second cup an empty package, and pour new sugar into the package until the scales are in equilibrium. The package will weigh 8kg.
• Put the kettlebell in the second cup, put into the first cup an empty package and pour out the sugar from the 8kg package into the package in the first cup until the scales are in equilibrium. The new package will weigh 4.5kg, therefore the first package now weighs 3.5kg.
• Put the 4.5kg package in the second cup, put into the first cup an empty package, and pour new sugar into the package until the scales are in equilibrium. The package will weigh 3.375kg.
• Put the 3.5kg package in the second cup, put into the first cup an empty package and pour out the sugar from the 3.375kg package into the package in the first cup until the scales are in equilibrium. The new package will weight 2.625kg, therefore the first package now weighs 0.75kg.
• Put the 0.75kg package in the first cup, put into the second cup an empty package, and pour new sugar into the package until the scales are in equilibrium. The package will weigh 1kg, as desired.

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My solution is the following:

Obviously it is possible...

This is why:

Weigh the kettlebell on one side and 8 on the other one (K_8)
Remove the kettlebell and weigh 6 Kg of sugar (6_8)
Remove the 8 and weigh K+2 (6_K+2)
Weigh 4.5 on one side and 6 on the other one (4.5_6)
Weigh other 4.5 (4.5_6)
Remove the 6 and weigh 4.5+1.5 (4.5_4.5+1.5)
Remove the 1.5 and weigh other 1.5 (4.5_4.5+1.5)
Remove 4.5 on the right side and put 2+1.5+1.5+1 (4.5_2+1.5+1.5+1)!

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