Here's another math trick that was used as part of a beverage commercial (7 UP): http://digicc.com/fido/

You can try that link or play with it here. I will give the answer for the first 4 replies.

1) Think of a 3 or 4 digit number and write it down.
2) Scramble the digits of that number.
3) Calculate the absolute value of the difference between the two numbers from steps #1 and #2.
4) Circle one of the digits in the difference. Don't circle zero. Report the remaining digits.
5) I will respond with the value of the digit you circled.

John

 

Ok John, the remaining digits are (in numerical order) are 468.

Dave

 

Dave, I believe you circled a 9. John

I am still testing my solution. The other option is 2.

 

Dave, I believe you circled a 9. John

Correct! And Fido got it right too!

Let me have a think about how this works. In the meantime lets try a 3-digit one: the remaining digit (in numerical order!) is 0.

Dave

 

9 John

My first solution to the puzzle seems correct.

John

 

9 John

My first solution to the puzzle seems correct.

John

Yep. I tried to be sneaky with that one by only doing a half-assed scramble.

Ok, last one before I crack on and have a bash at trying to work it out: the remaining digits (in numerical order!) are 036.

Dave

 

You must love the digit 9.
John

 

You must love the digit 9.
John

I do!

Ok, I tried to be sneaky again. It works alright, now to have a think about how it works. I've got some referencing to do later with a cool beer, ample time to have a muse over it!

Dave

 

There may be other solutions. I will give mine tomorrow. A cool beer always helps. John

 

Well, it has been a day. So, for those who didn't figure out the trick or who didn't try, here it is.

Those who went to school before the widespread use of adding machines often learned a method for checking addition, subtraction, and other arithmetic operations by "casting out nines." It is sort of like a check sum. It won't tell you the right answer, but it will tell you whether your answer is right. The solution to this puzzle is related to that procedure. Here is a Wikipedia article on the method: http://en.wikipedia.org/wiki/Casting_out_nines

Solution:

1) Add the digits in the remaining number and "cast out nines." Thus, if the sum of the digits is 17, then cast out a nine to get 8 as the remainder.
2) Subtract the remainder from 9 and the result is the number that was circled. That is, for a sum of digits of 17, the circled number was 1.
3) If the sum of digits is 9 or a multiple thereof, then the circled number was 9. Remember you cannot allow zero to be circled, or there are two solutions to sums of 0, 9, or any multiple of 9, namely 9 or 0.

I suspect this trick would work with any base, but I have not tried it. For hex, I guess it would be called casting out F's. Academically challenged students might find that distressing.

John

 

Very nice John. I had a think about it last night trying to pick some patterns and couldn't think of anything. Although it is quite some time since I came across the "cast out nines" method I would never have thought that was the method!

The absence of the option of circling zero should have been an indicator. Shame on me!

Dave

 

I might mention that there is an earlier post in the Math forum with a very similar trick.