The Trusses of Hay

'Farmer Tomkins had five trusses of hay, which he told his man Hodge to weigh before delivering them to a customer. The stupid fellow weighed them two at a time in all possible ways, and informed his master that the weights in pounds were 110, 112, 113, 114, 115, 116, 117, 118, 120, 121. Now, how was Farmer Tomkins to find out from these figures how much every one of the five trusses weighed singly? The reader may at first think that he ought to be told "which pair is which pair" or something of that sort, but it is quite unnecessary. Can you give the five correct weights?'

Extension: What is the smallest set (total weight) of five weights which, when taken in pairs, will produce 10 successive integral weights? Or can such a set be found? If not, find a set which produces integral weights, with as many successive integral weights as possible. (The above set has 7 successive integral weights, for example.)

Source: Henry Ernest Dudeny, in The Penguin Book of Curious and Interesting Puzzles, David Wells, 1992, #298. Used as Ken's POTD 10/25/94.

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