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Miscellaneous thoughts and ramblings
Wednesday, May 11, 2005
Think Clear
I promise I have no intention of turning this into an all-puzzle blog, but I love puzzles, and occasionally I find one I think you may enjoy.

My mother-in-law, Svenmom, sent me a link to a fun website that is an ad for 7up. (Click the title of this post to go there.) The site isn't a puzzle; it's a trick that "reads your mind". The puzzle is my challenge to you: explain how it works. I'll put up the answer in about 2 days.
Ok, they got my number but I don't know how.
Jack: Of course they got your number. The question is how? This takes a little math.

Oven: Where are you? This is right up your alley.

Psychotoddler: You went to med school. You gotta know some math. Blow off your next 5 patients and work on this.
If I get this right, it'll be a fluke. Math/deductive logic are not my thing.

Yes, it picked my number. I think the number it picks is a single digit version of both sets of original numbers I picked -- 3654 when added together make 18. 1+8=9.
My other number was 4563, which when added together make 18. 1+8=9.

When I did the subtraction, I got 909, circled the 9. 9+0=9, or even if I add all 3 digits, 9+0+9=18, reduced to a single digit = 9.

Okay, could Einstein have used me on his team!?
Torontopearl: You're too self-deprecating. You're actually on the right track. The sum of the digits is actually very relevant. Try some other numbers to test your hypotheisis. You can actually try numbers of any size (i.e. any number of digits).

You have the solution if you can do the same trick to someone else (i.e. they tell you the rest of the digits and you tell them the number they circled).
By G-d, I think HE's got it! My husband solved it in two minutes -- that's why he's an accountant and I'm not.

"Whatever result you get after subtracting one set of digits from the other, the sum of the "result" will ALWAYS be 18. If you know that, then you find the difference between the numbers submitted and 18."
Oops, my husband apologizes. He should have taken a larger sampling of equations instead of a random three that kept totaling 18. At least it's a lucky 18!
We're back...
After further review,hubby noticed a pattern. The sum of all the digits will be divisible by 9 -- the answers will be either 9, 18, or 27.
After taking one number away and you add up the remaining numbers, if the answer is less than 9, then you subtract the answer from 9 to get the circled number. If the answer is between 9 and 17, then you subtract the answer from 18 to get the circled number. If the answer is between 18 and 26, then you subtract the answer from 27 to get the circled number, etc.

I think he really deserves a coffee and a stale muffin right about now!
Torontopearl: Very good. The sum of the digits of the number you get after you subtract the second number from the first will always be divisible by 9. So when one number is taken away, it can be determined by simply finding the number that is needed to add to the sum of the digits to make a number that's divisible by 9.

For you math geeks:
circled number = 9 - [ sum of other digits mod 9]
where mod is the modulo operator which gives you the remainder when the first operand is divided by the second. OK?

Excellent work, Torontopearl's husband. You will find a stale croissant and some decaf in your floppy drive right now. Sorry, we're out of muffins.

Some more mysteries remain:
1) How is it that taking any number and scrambling the digits and subtracting one number from the other gives a number whose digits sum to a multiple of 9?
2) Is there something more general we can say about numbers whose digits sum to a multiple of 9?
3) Can you figure out now why the website won't let you circle a zero?

Answers to these questions will earn a half double-caf half decaf espresso with a twist of lemon from the USB port of your choice.
Hubby thanks you!(it isn't called a "brain teaser" for nothing)
I don't think I'll have time to post the answers today, so look for them Sunday. That still gives the intrepid math lovers out there another couple of days to answer my questions above and earn many kudos and an entirely fictional cup of joe.
I may be wrong, but I think this involves casting nines and modular arithmetic.

I don't think the first couple steps of the puzzle matter as long as you don't choose a number with too many zeros or a number without integer variations, i.e., 999, 888, etc. (That's because 999-999=0.) The sum of 0 = 0. C

Getting to this is a little complicated, so I'll just explain the end result (without giving the explanation of how to get there).

Add together the sum of the numbers you type in at the end of the puzzle, i.e., lets say my number was 1053. If I type in 153, that's 1+5+3 = 9. The sum is 9, so the missing digit is 0.

If it's 0, the missing digit had to be 9.
If it's 1, the missing digit had to be 8.
If it's 2, the missing digit had to be 7.
If it's 3, the missing digit had to be 6.
If it's 4, the missing digit had to be 5.
If it's 5, the missing digit had to be 4.
If it's 6, the missing digit had to be 3.
If it's 7, the missing digit had to be 2.
If it's 8, the missing digit had to be 1.
If it's 9, the missing digit had to be 0.

Am I right? If you think puzzles like this are fun, you'd like my family!
The first and last of the list don't always hold true.
Voracious Reader: Yup, but you're stating what Torontopearl's husband has already discovered. (Take a look at her comments above.)

Here's the rest of the solution.

First of all, a number is divisible by 9 if and only if the sum of its digits is divisible by 9. (That makes for an easy way to quickly check if a large number is divisible by 9.) Now, taking any integer (of any number of digits) scrambling the digits, and subtracting one from the other, always yields a number that's divisible by 9. The rest is basically what Mr. Torontopearl figured out.

The reason you can't circle a zero is because if the sum of the remaining digits is divisible by 9, there's no way to determine if the circled digit is a zero or a 9, so one of the two has to be eliminated as a possibility.
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