Calendar PuzzleDate: 09/09/2001 at 23:56:35 From: Bob Stoddard Subject: Puzzle/trick My teacher showed us a calendar with five weeks S M T W T F S in it, nothing special, he asked us to choose 1 2 3 4 only one day out of each week and write what 5 6 7 8 9 10 11 the date was of each day. The he told us to add 12 13 14 15 16 17 18 up all the dates. 19 20 21 22 23 24 25 26 27 28 29 30 I chose the days: Friday the 3rd Wednesday the 8th Monday the 13 Tuesday the 21 Tuesday the 28 ... which equals 73. Then he wrote all the days of the week on the board, like this: S M T W T F S and asked what days I chose, NOT the dates, and the marked them under the days of the week diagram he made on the board. Somehow he knew what the total of all the dates added up to, without looking at the calendar - just looking at the diagram he made. I just want to know how he did it. I've tried to figure it out but it's quite the brain teaser. Thanks. Sincerely, Bob Stoddard Date: 09/10/2001 at 13:05:22 From: Doctor Peterson Subject: Re: Puzzle/trick Hi, Bob. The key is that you had to choose one date from each week. For a hint as to how it works, suppose that you had swapped the weeks from which you chose your Friday and Wednesday, so that they were the 10th and 1st rather than the 3rd and 8th. Do you notice that the sum of these two is still 11? It turns out that no matter how you rearrange the weeks from which you chose each date, while using each day of the week the same number of times, the sum doesn't change! So he can find the sum just by making his own choice that matches, like this: S M T W T F S 1 2 (3) 4 5 6 7 (8) 9 10 11 12 13(14)15 16 17 18 19 20(21)22 23 24 25 26(27)28 29 30 1 2 1 1 -- -- -- -- 27+14+ 8 + 3 +21 = 73 There may be a more specific method he used to get the sum quickly; I'll suggest one at the end. But this is enough to make the trick work. Now, let's do this more precisely. To make it easier, I'm going to extend the calendar so all the rows are full: S M T W T F S 0: -2 -1 0 1 2 3 4 1: 5 6 7 8 9 10 11 2: 12 13 14 15 16 17 18 3: 19 20 21 22 23 24 25 4: 26 27 28 29 30 31 32 (It doesn't matter that you aren't going to choose any of the dates I've added.) I've also marked week numbers, starting with zero. We'll use the number of the day at the top of each column to represent that day of the week; for example, the 13th is day -1 (Monday) of week 2 (the third week). Got it? Now the date for any day D of any week W will be D + 7W That is, to find the date, you just start with D, the number at the top of the column, and add 7 for each week you move down, so the date is the day number plus seven times the week number. Now you pick five dates, each with its own day number (which is up to you), but one from each week (so your list must include all 5 possible values for W): D1 + 7*0 D2 + 7*1 D3 + 7*2 D4 + 7*3 D5 + 7*4 Add these up, and the sum is the sum of your five day numbers, plus the sum 0+7+14+21+28 = 70. (This is also 7 times 0+1+2+3+4.) So to find the sum of your dates, all I have to do is to add up the dates at the top of the columns you chose, and add 70. That's the easy way to find the total; it also proves that the total depends only on which columns you chose from. In your example, M+T+T+W+F = -1+0+0+1+3 = 3; add 70 and we get 73. Can you see what you have to do differently if the month you choose happens to have 4 or 6 weeks instead of 5? Just for fun, you might try doing the trick with your teacher using such a calendar, and see if he knows how to handle it. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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