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Best of Donald Catlin

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A Slick Proposition Bet

5 December 2008

Every now and then I write about proposition bets. Here is a dandy.

You ask your mark to write down any four-digit number and not let you see it. Then tell him to rearrange (permute) the numbers and again not let you see what is being done. Now have him subtract the smaller number from the larger. Now have him circle one of the numbers other than zero and then rearrange the remaining three numbers. All of this is done without your being privy to his actions. Have him tell you what the final three digit number is; this is the only information you have. Then bet him that you can guess the number that he circled. Sound impossible? Well, before you read any further do the following.

Go to your favorite search engine and type in Fido Puzzle. The first entry listed will probably be the site you want. If you go to it you'll have a puzzle based on the above proposition. Try it a few times. (Warning, the box where you enter your final number has no cursor so just type and your final number will appear.) Amazing isn't it? And, no, I don't know why the word "Fido" is used.

Here's the scoop. Suppose that the number is a,b,c,d. Suppose that the rearranged number is subtracted from the starting number, although, as you'll see, it really doesn't matter. Now in at least one case you'll have to borrow a one from the lefthand neighbor to carry out the subtraction. Suppose that d borrows from c. Then the number looks like a,b,c-1,d+10. If we add these digits we have a + b + c + d + 9. When we subtract the rearranged digits and sum the result the total will be 9. Suppose, however, that c-1 has to borrow from b. Then the number would look like a,b-1,c+9,d+10. Here the digits sum up to 18 so when we subtract our rearranged number the digits in the result add up to 18. I think you get the idea. The operation of subtracting the smaller from the larger leaves you with a number whose digits add up to a multiple of 9. Suppose this number is e,f,g,h. If you circle g, for example, then the remaining numbers, however you rearrange them, will have a total less than some multiple of 9 (this is why zero is ruled out). Hence subtract e + f + h from the next multiple of 9 and the result will be g.

Let's do an example. Suppose the original number was 8463. I rearrange it as 3486. 8463 - 3486 = 4977. Notice the digits add up to 27. Suppose that I circle 4 and rearrange the rest as 797. These digits add up to 23. Subtracting 23 from 27 leaves a total of 4. Cute!

See you next month and I'll get back to casino games.


Don Catlin can be reached at 711cat@comcast.net

Donald Catlin

Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers
Donald Catlin
Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers