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August 23rd, 2011, 01:56 AM   #1
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Trick / Pr(Digital Root of 9 occuring)

...So i wanted to find a trick where i could use digital roots to solve the problem, and since their properties are very much unheard of no-one would be able to guess how i had done it. However i wanted to also be able to fool those who did know about digital roots with a trick which seemed impossible!
"Choose 10 irrational numbers and multiply them all together" -> "Now, tell me all the digits in the product that your left with apart from the first one" -> "The missing digit was: (9 - the digital root of that number)!!!!"

For my trick to work though it required probability being in my favor. But since the probability of the final product having a digital root of 9 increases as the number of irrational numbers involved in the multiplication increases, i decided to find a method for quickly working out all the probabilities for the digital root of 9, so i could compare which

Where X is the number of irrational numbers involved in the multiplication and Y is the probability of the digital root being 9,
Pr(Y)= 1-[(2/3)^X + (2X/3)*(2/3)^(X-1)]
Therefore with 4 numbers multiplied together, the trick will work 53.9% of the time, and with 10 numbers, it will work with a 92.48% chance.

All this being said i wasn't finished. I realised that if the number already started with a digital root of 3 or 6, most people wouldn't realise what the trick was using, and i could increase the chance, or better still decrease X . (tell them to multiply their first number by something with a digital root of 6 or 3)
The rule was: Pr(Y|is a digital root of 3 or 6)= 1-(2/3)^X

Now with the new equation i could have 4 numbers multiplied together, and the trick will work 80.25% of the time, and with 10 numbers, it would now work with a 98.27% chance of success!

And if you just want 100% success rate on this trick tell them to pick one irrational number and to multiply it by something with a digital root of 9
So if you know someone smart and want to mess with their head try using this trick on them!
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August 23rd, 2011, 06:17 AM   #2
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Re: Trick / Pr(Digital Root of 9 occuring)

Neat trick. I'm not sure why you say irrational numbers, though; I would think you'd want positive integers. (Most irrational numbers have product of digits = 0.)
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August 26th, 2011, 03:48 AM   #3
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Re: Trick / Pr(Digital Root of 9 occuring)

Quote:
Originally Posted by CRGreathouse
(Most irrational numbers have product of digits = 0.)
hmmm...only 0 has the digital root of 0 nothing else, unless your using the method "discard the nines" you cannot reach 0 as the digital root by simply adding all the digits of a number.
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August 26th, 2011, 05:39 AM   #4
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Re: Trick / Pr(Digital Root of 9 occuring)

I'm working mod 9, which is equivalent to the digital root except in the case of 0. Zero mod nine is the same as "divisible by nine", and both are the same as digital root nine for nonzero numbers.
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