Calendar Puzzle

Date: 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/