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The answer is Because As such Note that with $9$ students this would be: In general for squares of side $n$ these are the ...and for rectangles of sides $m,n$ the result has

This may not yet be isn't quite the best...

I can, using a method devised below, do it in Starting with a simple case, 1_0, it's pretty obvious the optimal steps are: 1_0 1_0 _10 or 10_ 01_ _01 0_1 0_1 both of whi …

In the language of combinatorics the number of symbols, assuming only purely rotational and reflectional geometric symmetry* is: * as pointed out by Ian MacDonald and Silenus, this has not been guaranteed …

For what it's worth

The answer is Because I used this Python code: Reduced bounds from before I started a search... Because A lower value for the upper bound is: Lowered further by @astralfenix to …

The board Because Furthermore

An upper bound is Note that Now we can Some code: def squarePairs(n): for a in range(1, n + 1): for b in range(a + 1, n + 1): if ((a + b) ** .5) % 1 == 0: …

I wrote a program to search for such dice. It found $29$ with maximum faces of It finds none with maximum faces of less. Here are those dice: The program works by first finding all ($71$ after …

I just had a fun visit from a time traveller and she offered to whisk me off for a whirlwind tour – she offered to take me to observe any one time for each of any choice of distinct years from $1$C.E. …

If we also know that all the children are residents then there are There are $80$ residents at home All residents have the same probability, $p$, of taking the elevator at a certain time (eith …

Without allowing backtracking along edges Since The paths

I will put this out there, since enumeration seems to hold, although I can't quite see how it's the answer yet (i.e. what the base choices actually are that lead to this result). What is this? . …

Here is my generic solution. Code to perform $_nC_k$: def choose(n, k): '''Returns the number of ways to choose k items from n items''' reflect = n - k if k > reflect: if k > n: …

As you say, you are implementing a type of Douglas Engel's puzzle, an "Enigma". Given the colour set and the fact that you are ignoring the orientation of the triangular wedges ("corners" in twisty pu …