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I have recently found the great card came - SET. Briefly, there are 81 cards with the four features: symbol (oval, squiggle or diamond), color (red, purple or green), number (one, two or three) and shading (solid, striped or open). The task is to locate (from selected 12 cards) a SET of 3 cards, in which each of the four features is either all the same on each card or all different on each card (no 2+1 combination).
I've coded it in MATLAB to find a solution and to estimate odds of having a set in randomly selected cards.
Here is my code to estimate odds:
%% initialization
K = 12; % cards to draw
NF = 4; % number of features (usually 3 or 4)
setallcards = unique(nchoosek(repmat(1:3,1,NF),NF),'rows'); % all cards: rows - cards, columns - features
setallcomb = nchoosek(1:K,3); % index of all combinations of K cards by 3
%% test
tic
NIter=1e2; % number of test iterations
setexists = 0; % test results holder
% C = progress('init'); % if you have progress function from FileExchange
for d = 1:NIter
% C = progress(C,d/NIter);
% cards for current test
setdrawncardidx = randi(size(setallcards,1),K,1);
setdrawncards = setallcards(setdrawncardidx,:);
% find all sets in current test iteration
for setcombidx = 1:size(setallcomb,1)
setcomb = setdrawncards(setallcomb(setcombidx,:),:);
if all(arrayfun(@(x) numel(unique(setcomb(:,x))), 1:NF)~=2) % test one combination
setexists = setexists + 1;
break % to find only the first set
end
end
end
fprintf('Set:NoSet = %g:%g = %g:1\n', setexists, NIter-setexists, setexists/(NIter-setexists))
toc
100-1000 iterations are fast, but be careful with more. One million iterations takes about 15 hours on my home computer. Anyway, with 12 cards and 4 features I've got around 13:1 of having a set. This is actually a problem. The instruction book said this number should be 33:1. And it was recently confirmed by Peter Norvig. He provides the Python code, but I didn't test it yet.
So can you find an error? Any comments on performance improvement are welcome.