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        riddles >> easy >> A mathematical trick by Fido
        
(Message started by: bele on Sep 25^th, 2003, 1:31pm)

Title: A mathematical trick by Fido
Post by bele on Sep 25^th, 2003, 1:31pm

Hi everyone,

I wonder whether anyone will be able to help me solve this problem. I came across this website where you're asked to choose a random 4 digit number..... Instead of explaining it, please go to the following website to see how they can guess the number you choose.

http://digicc.com/fido/

Is there a formula which can be applied to solve the problem? I really like to know how they get the answer.

Thanking everyone in advance.

Bele

Title: Re: A mathematical trick by Fido
Post by James Fingas on Sep 25^th, 2003, 1:53pm

He tells you not to pick a zero, but if you do, he thinks you picked a 9. In fact, if you enter all of the digits, he thinks you picked a 9 as well. He tells you to pick a number with lots of different digits so you don't get all zeros after you subtract.

You could also try entering test numbers like "1", "01", "10", "100000", "2", "3", "4", "1111", etc.

Title: Re: A mathematical trick by Fido
Post by bele on Sep 25^th, 2003, 2:43pm

Hi,

But I still don't understand how he knows what number I picked. There must be some logic or trick. It could be better if some examples were used to explain how the trick works.

Thanks.

Bele

Title: Re: A mathematical trick by Fido
Post by Sir Col on Sep 25^th, 2003, 3:34pm

Hi Bele! I won't give it away immediately, but I'll provide three hidden hints which sequentially lead you towards the trick...

Hint 1:
::[hide]
When you find the difference between a number and its reverse, the answer is always divisible by 9. Proving this in general is tricky, but it's relatively easy for a 4-digit number (abcd). You should get the difference to be: 9[111(a–d)+10(b–c)].
E.g. Suppose we pick 3621. Difference = 3621–1263 = 2358 = 9x262
[/hide]::

Hint 2:
::[hide]
If a number is divisible by 9, so too is the sum of its digits.
In my example, 2+3+5+8 = 18.
[/hide]::

Hint 3:
::[hide]
Suppose I give you three digits of my number. They will add to a total a little short of a number in the 9 times table.
In my example (2358), I might remove, say 2. So the remaining digits add to 3+5+8=16. The next number in the 9 times table is 18, so I know the number missing (circled) is 2.
[/hide]::

It's a variation of an old trick, but it was presented superbly. Thanks for sharing the link with us, Bele.

Title: Re: A mathematical trick by Fido
Post by Icarus on Sep 25^th, 2003, 4:15pm

Practically all these "I'll tell you what number you picked" type problems rely on variations of the same trick. I don't know your current level of mathematical knowledge, but if you are comfortable with some simple algebra, try this: let a, b, c, and d be the four digits of the number you start with. Write your number out as:

a*1000 + b*100 + c*10 + d.
Now choose any rearrangement
of the digits: say (bdca) instead of (abcd). The corresponding number is

b*1000 + d*100 + c*10 + a.

Now take the difference like he tells you:

(abcd) - (bdca) = a*1000 + b*100 + c*10 + d - (b*1000 + d*100 + c*10 + a)
= a*1000 + b*100 + c*10 + d - b*1000 - d*100 - c*10 - a
= (a*1000 - a) + (b*100 - b*1000) + (c*10 - c*10) + (d - d*100)
= 9999*a + 900*b + 0*c - 99*d.

Dol you notice something about these numbers? Try it with other arrangements, you will see the same thing always happens.

Now put that aside for a minute. Consider again an arbitrary number. This works for any number of digits, but let's look at four digits again:

(abcd) - (a + b + c + d) = a*1000 + b*100 + c*10 + d - a - b - c - d = a*999 + b*99 + c*9 + d*0.

Thus, the difference between any number and any other number resulting from a rearrangement of its digits is always divisible by 9. And the difference between any number and the sum of its digits is also always divisible by 9.

Put these two facts together, and you should be able to figure out how he does it, and also why he doesn't want you to choose zero as your digit.

Feh - that's what happens when you take the time to write it out carefully (and stop for supper in the middle). Some upstart Brit Math teacher jumps in with the same info ahead of you! >:(

Title: Re: A mathematical trick by Fido
Post by bele on Sep 25^th, 2003, 7:59pm

Hi Sir Col and Icarus,

Thanks for the prompt reply. I really appreciate your efforts in explaining the solution with examples as well as with formulas. It makes it doubly clear. Once again thank you.

Take care.

Bele

Title: Re: A mathematical trick by Fido
Post by Sir Col on Sep 26^th, 2003, 4:33pm

on 09/25/03 at 16:15:27, Icarus wrote:

Some upstart Brit Math teacher jumps in with the same info ahead of you! >:(

Upstart? Huh! And, English, please. ;)

I'd never really thought about what you said, Icarus: the difference between a natural number and any random configuration of its digits is divisible by 9. Of course, it follows from the standard proof that the difference between any natural number and its reverse is a multiple of 9. Nice! Thanks for sharing that insight.

Title: Re: A mathematical trick by Fido
Post by Icarus on Sep 27^th, 2003, 6:41pm

Though I vaguely remember hearing it before, I never really thought about it either. But it was readily apparent (to those of us with more mathematics education than we know what to do with, Bele ;)) that this was required for the trick to work, and once you think about it, it is easy to prove.