Patterns and Puzzles at CTY

For the next few weeks, I will be at Johns Hopkins Center for Talented Youth Program studying something called "Inductive and Deductive Reasoning." I will be posting anything I learn during the week that would appeal to you.

We learned some really cool patterns that you guys will definitely like. The first of which involves just ones. Take 1 x 1.

1 x 1 = 1

Now, try 11 x 11

11 x 11 = 121

How about 111 x 111

111 x 111 = 12321

Let's try them all the way through to 9 ones. Pull out your calculators and you should see that it does. What about with ten ones? Sure enough, the pattern does continue, but in disguise. Try opening up a spreadsheet (unless you own a 19+ digit calculator) and type it in. You should get 1234567900987654321. This is because the 10 couldn't fit in that spot, so the zero dropped in and the one carried over to the nine which carried to the eight, giving you this answer. And this pattern I believe continues (if you find a proof, please inform me of it or post it for us) forever just by carrying ones and what not.

Also, let's try taking these sequences of 123456... How about we go on and stop. Then, we multiply by eight and add the number we stopped at. Let's see:

1 x 8 + 1 = 9
12 x 8 + 2 = 98
123 x 8 + 3 = 987
1234 x 8 + 4 = 9876
12345 x 8 + 5 = 98765
123456 x 8 + 6 = 987654

This pattern continues all the way up to 123456789. I'm not sure what happens after that, but it's still really cool. If you have a proof of this, please post it!

Bonus: In class, we've done a few puzzles. I would like to share one I really liked with you.

You are by a river with only a nine quart bucket and a four quart bucket (ONLY that, no bucket to dump it in at the end), and need to bring exactly six quarts of water back from the river. Following these guidelines, how would you get the water?

Email me the answer you get and I will tell you if it's right. If you want the solution, I will tell you, but don't spoil it for the others!!

Problem of the Week solution (from June):

Easy Problem:
c = 5
z = 3
n = 6
y = 1
odds = 71.4 %

Hard Problem:
a = 31.2 in
b = 24.1 in
x = 1
y = 3
m = -3
r = 6
n = -1
g = 5/3
q = 5.2
p = 9.5
z = 12.3
h = 8.2
area = 89.38 or 89.4 cm^2