101. THE TRUSSES OF HAY. Farmer Tompkins 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,
and 121. Now, how was Farmer Tompkins to find out from these figures how much every one of the five trusses weighed singly? The reader may at (1/2)
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? (2/2)
SOLUTION TO 101. THE TRUSSES OF HAY. (1/2)
Add together the ten weights and divide by 4, and we get 289 lbs. as the weight of the five trusses together. If we call the five trusses in the order of weight A, B, C, D, and E, the lightest being A and the heaviest E, then the lightest, no lbs., must be the weight of A and B; and the next lightest, 112 lbs., must be the weight of A and C. Then the two heaviest, D and E, must weigh 121 lbs., and C and E must weigh 120 lbs.
SOLUTION TO 101. THE TRUSSES OF HAY. (2/2)
We thus know that A, B, D, and E weigh together 231 lbs., which,
deducted from 289 lbs. (the weight of the five trusses), gives us the weight of C as 58 lbs. Now, by mere subtraction, we find the weight of each of the five trusses--54 lbs., 56 lbs., 58 lbs., 59 lbs., and 62 lbs. respectively.
SOLUTION TO 232. CATCHING THE MICE. (1/6)
In order that the cat should eat every thirteenth mouse, and the white mouse last of all, it is necessary that the count should begin at the seventh mouse (calling the white one the first)--that is, at the one nearest the tip of the cat's tail. In this case it is not at all necessary to try starting at all the mice in turn until you come to the right one, for you can just start anywhere and note how far distant the
SOLUTION TO 232. CATCHING THE MICE. (2/6)
last one eaten is from the starting point. You will find it to be the eighth, and therefore must start at the eighth, counting backwards from the white mouse. This is the one I have indicated.
In the case of the second puzzle, where you have to find the smallest number with which the cat may start at the white mouse and eat this one last of all, unless you have mastered the general solution of the
SOLUTION TO 232. CATCHING THE MICE. (3/6)
problem, which is very difficult, there is no better course open to you than to try every number in succession until you come to one that works correctly. The smallest number is twenty-one. If you have to proceed by trial, you will shorten your labour a great deal by only counting out the remainders when the number is divided successively by 13, 12, 11,
10, etc. Thus, in the case of 21, we have the remainders 8, 9, 10, 1, 3,
SOLUTION TO 232. CATCHING THE MICE. (4/6)
5, 7, 3, 1, 1, 3, 1, 1. Note that I do not give the remainders of 7, 3,
and 1 as nought, but as 7, 3, and 1. Now, count round each of these numbers in turn, and you will find that the white mouse is killed last of all. Of course, if we wanted simply any number, not the smallest, the solution is very easy, for we merely take the least common multiple of 13, 12, 11, 10, etc. down to 2. This is 360360, and you will find that
SOLUTION TO 232. CATCHING THE MICE. (5/6)
the first count kills the thirteenth mouse, the next the twelfth, the next the eleventh, and so on down to the first. But the most arithmetically inclined cat could not be expected to take such a big number when a small one like twenty-one would equally serve its purpose.
In the third case, the smallest number is 100. The number 1,000 would also do, and there are just seventy-two other numbers between these that
SOLUTION TO 232. CATCHING THE MICE. (6/6)
the cat might employ with equal success.
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