Parlour Magician

Number Theory Level 4

In a parlour game, the “magician” asks one of the participants to
think of a three-digit number ${ abc }_{ 10 }$ . Then the magician asks the participant to add the five numbers ${ acb }_{ 10 }$ , ${ bca }_{ 10 }$ , ${ bac }_{ 10 }$ , ${ cab }_{ 10 }$ and ${ cba }_{ 10 }$ , and reveal their sum. Suppose the sum was $3194$ . What was ${ abc }_{ 10 }$ originally?

Details and Assumptions

${ abc }_{ 10 }$ means a 3-digit number with $a,b,c$ as it's digits and not $a\times b\times c$ .

This problem is not original.

The answer is 358.

2 solutions

Rahul Dandwate
Oct 12, 2014

The sum of the 5 numbers can be written as

$222(a+b+c)-{ abc }_{ 10 }=3194$

i.e.

$222(a+b+c-14)-86={ abc }_{ 20 }$

which clearly gives us

$a+b+c>14$

And,

$222(a+b+c-14)-86<1000$

That gives $(a+b+c)\in \{ 15,16,17,18\}$

Trying out these values of $a+b+c$ will give the answer to be $\boxed{358}$ .

Nafees Zakir
Oct 12, 2014

>>> for a in range(0,10):
    for b in range(0,10):
        for c in range(0,10):
            total = (a*100+c*10+b)+(b*100+c*10+a)+(c*100+a*10+b)+(b*100+a*10+c)+(c*100+b*10+a)
            if(total==3194):
                print "%d%d%d" % (a,b,c)


358

So the answer would be $\boxed{358}$

Parlour Magician

The answer is 358.

2 solutions

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