Cynthia Lanius
Algebra - Fun with Calendars
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A fun mathematical puzzle to play with your friends.
(Or teachers, with your class.)
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18 |
19 |
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25 |
26 |
Take any
calendar. Tell your friend to choose 4 days that form a square like the four to
the right. Your friend should tell you only the sum of the four days, and you
can tell her what the four days are.
How does the puzzle work? You
know how people always want to see a use for algebra? Well this puzzle uses algebra. Here's what I mean.
Let's pretend that the 4
numbers that the person chose were the highlighted ones here - 18, 19, 25, and
26. She adds up the four numbers and tells you only that the sum is 88.
You make a couple of
calculations and tell her the numbers. What calculations? Lets figure that out
with algebra. Let's call the first number n. Then you know that
the next number would be n + 1 and the next number would be n
+ 7 and the next number would be n + 8. We had our friend
add up the four numbers, so let's add our four numbers:
n + n + 1 + n + 7 + n + 8
And since our friend got 88
when she added them, let's make our sum equal 88:
n + n + 1 + n + 7 + n + 8 = 88
Simplify our equation by
adding like terms:
4n + 16 = 88
How would you solve this
equation? Subtract 16 from both sides?
4n = 72
Divide both sides by 4?
n = 18
Subtract 16 and divide
by 4. That's
exactly how you solve the puzzle. When your friend tells you the sum, you
subtract 16 then divide by 4. This gives you the first number n.
(Then add 1 and 7 and 8 for the other numbers).
Alternate and easier
method: Subtracting
16 mentally isn't so easy. Go back to that equation:
4n + 16 = 88
I think I see a better way.
Factor 4 from the left side of the equation:
4(n + 4) = 88
Now, I could divide both
sides by 4:
(n + 4) = 22
Subtract 4 from both sides.
n = 18
That's a lot easier
to do mentally. Divide by four and then subtract 4.
Summary: So how does the puzzle work again? Your
friend adds any 4 numbers that form a square on the calendar and tells you the
sum. You divide by four and then subtract 4. That gives you the first
number. You add 1, 7, and 8 to get the other numbers.
And algebra makes it all
possible.
Other Math Lessons by Cynthia Lanius
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Copyright 1997-2008 by Cynthia
Lanius.
URL
http://math.rice.edu/~lanius/Lessons/calen.html