A forklift only starts if the total load is exactly 100 kg or exactly 150 kg. You have 5 indivisible boxes of unknown weights. An operator left these notes: - any 3 boxes together weigh less than 100 kg; any 4 boxes together weigh more than 150 kg; there are at least two different pairs that weigh exactly 100 kg. Does it necessarily follow that there is some trio that weighs exactly 150 kg?

No. There is no selection that adds exactly 100 kg or exactly 150 kg.

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The combined weight lift

The combined weight elevator works when the reader accepts a simple rule and follows it to the end without losing the thread. That's where the beautiful part appears: in the clarity of the mechanism.

A forklift only starts if the total load is exactly 100 kg or exactly 150 kg. You have 5 indivisible boxes of unknown weights. An operator left these notes: - any 3 boxes together weigh less than 100 kg;

any 4 boxes together weigh more than 150 kg;
there are at least two different pairs that weigh exactly 100 kg. Does it necessarily follow that there is some trio that weighs exactly 150 kg?

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Hints

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It is not necessary to know each box separately: the sums in pairs already contain a lot of information.
Try to recover the total sum of the five boxes from the ten sums of pairs.
If you know that total sum, you can also deduce the weight of each box by subtracting the sum of the four pairs that do not contain it.

Solution

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Answer: No there is no selection of boxes that add up to exactly 100 kg or 150 kg. Explanation: Call the weights of the five boxes $a,b,c,d,e$. The sum of all the pairs is

$$ 110+112+113+114+115+116+117+118+120+121=1156. $$ But each box appears in exactly 4 of those pairs, so $$

4(a+b+c+d+e)=1156,

$$ and therefore $$

a+b+c+d+e=289.
$$ Now order the pairs. The difference between consecutive sums allows the individual weights to be reconstructed, except in order. The set turns out to be

$$ 54,\ 56,\ 57,\ 58,\ 64. $$ With those five weights you can verify that: - no box weighs 100 or 150; - no couple adds 100 or 150; - no shortlist adds up to 100 or 150; - and, since the total is 289, neither do the quaternas nor the five together. Then the forklift cannot be activated with any selection. $$

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