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Dicey numbers (Posted on 2009-03-27) Difficulty: 3 of 5
A small number of standard dice were thrown. The total of the spots on the tops was a perfect square. Then the digits that represent each of the top numbers were placed in a row in increasing order to spell out a rather large number (a given digit may appear more than once, as in 12224).

The same thing was repeated with twice as many dice and again the total number of pips was a perfect square and a number formed by using the digits that represent the pips in ascending order was the square of the first number formed that way.

What were the two numbers formed by the digits representing the pips (the first number and its square)?

See The Solution Submitted by Charlie    
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Generalized solution Comment 6 of 6 |

continuing from my previous post I need both 3n+4 and 6n+7 to be squares thus I wanted

3n+4=y^2 and 6n+7=x^2 so we have

n=(y^2-4)/3 and thus

6(y^2-4)/3 + 7=x^2



and this is simply Pell's equation for D=-2 and thus we can have a generalized solution with

x(1)=7 y(1)=5 and then

x(t+1)=3x(t)+4y(t)   y(t+1)=2x(t)+3y(t)

and then simply taking N=[x(t)-7]/6 gives us the general solution where N is used to generate the solution with N 3's followed by a 4.

Now I am not entirely certain that these are the only solutions.

  Posted by Daniel on 2009-03-29 03:10:33
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