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**The problem**

For the first annulus, A(1), with radii r(1) and r(2), n=4 is the minimum value of n such that a regular n-gon lies wholly within or on the boundary of A(1).

Find the smallest value of n, such that a regular n-gon will not fall outside the annulus A(m) bounded by r(m) and r(m+1) for values of m = 2, 3 and 4.

Also, what is the smallest value of n for A(50)?

Die P has the numbers 3, 3, 3, 3, 3, 6.

Die Q has the numbers 2, 2, 2, 5, 5, 5.

The game itself is very simple: whoever rolls the highest total wins a sweet from the candy box. If the box becomes empty, the winner can take one from the other player’s pile.

The question is to find the smallest value of N, such that the following equation holds true:

4N = 4*N4

Again, using the example above, this would require that 4123 = 4*1234. This is obviously not true, so N=123 is not a solution.

So, find the smallest value of N and its length n.

If that was too easy, can we generalise the problem? Let K be our single-digit number, between 1 and 9 inclusive, such that KN = K*NK has a solution number N.

*two*problems; similar but different.

The solution has three steps: firstly, we need to find the relationship between the three radii; then there is a bit of number theory to find explicit numbers for

*a*and

*b*; and finally, some trigonometry to calculate the area of the large triangle.

The c=4 case