## Project Euler Problem 285

Another quite nice problem from Project Euler is number 285. The result of the problem depends on the computation of a certain probability, which in turn is related to the computation of a certain area. Below is an illustration of the computation for k equal to 10.

To save you some time, here’s a picture of the case k=1 which I ignored and spent quite a lot of time debugging… Plus, it only affects the last three digits or so after the decimal point…

Here’s a Matlab code which can construct the pictures above and can compute the result for low cases. To solve the problem, you should compute explicitly all these areas.

function problem285(k) N = 100000; a = rand(1,N); b = rand(1,N); ind = find(abs(sqrt((k*a+1).^2+(k*b+1).^2)-k)<0.5); plot(a(ind),b(ind),'.'); axis equal M = k; pl = 1; for k=1:M if mod(k,100)==0 k end r1 = (k+0.5)/k; r2 = (k-0.5)/k; f1 = @(x) (x<=(-1/k+r1)).*(x>=(-1/k-r1)).*(sqrt(r1^2-(x+1/k).^2)-1/k).*(x>=0).*(x<=1); f1 = @(x) f1(x).*(f1(x)>=0); f2 = @(x) (x<=(-1/k+r2)).*(x>=(-1/k-r2)).*(sqrt(r2^2-(x+1/k).^2)-1/k).*(x>=0).*(x<=1); f2 = @(x) f2(x).*(f2(x)>=0); if k == pl thetas = linspace(0,pi/2,200); hold on plot(-1/k+r1*cos(thetas),-1/k+r1*sin(thetas),'r','LineWidth',2); plot(-1/k+r2*cos(thetas),-1/k+r2*sin(thetas),'r','LineWidth',2); plot([0 1 1 0 0],[0 0 1 1 0],'k','LineWidth',3); hold off axis off end A(k) = integral(@(x) f1(x)-f2(x),0,1); end xs = xlim; ys = ylim; w = 0.01; axis([xs(1)-w xs(2)+w ys(1)-w ys(2)+w]); sum((1:k).*A)

## Two more mind games solved using Matlab

After doing this, solving the Sudoku puzzle with Matlab, I wondered what else can we do using integer programming? Many of the mind games you can find on the Internet can be modelled as matrix problems with linear constraints. I will present two of them, as well as their solutions, below. The game ideas are taken from www.brainbashers.com. Note that in order to use the pieces of code I wrote you need to install the YALMIP toolbox.

You have a table and on each row and column you need to have squares coloured with color and squares coloured with color such that no three squares taken vertically or horizontally have the same color. In the begining you are given the colors of some of the square such that it leads to a unique solution satisfying the given properties. The goal is to find the colors corresponding to each small square starting from the ones given such that the end configuration satisfies the given properties.