The link to the eight queens puzzle, however, is a good start for solving your problem on any board (go down to the "Exercise in algorithm design" section).
For instance, you can use a recursive algorithm to find all the ways to place p pawns at least distance D apart on a given board, by...
I'm afraid the ideas in the previous post won't be too helpful in solving your problem for D>1 (for D=1 with p pawns and any board with n spaces the answer is n choose p), but what will do the trick is setting up a recurrence relation with initial conditions and either solving it in general or...
This is not the matrix I described above, but it should be obvious that continuing the construction below for an infinite matrix will make each row sum -1 and each column sum 1.
\begin{array}{cccccccccc}
1 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 &...
Consider the infinite matrix where for each nonnegative k and each nonnegative j less than 2^k, row j+2^k has entry 1 in column k+1 and entry -1.5 in column k+2 and entry 0 elsewhere. Then each row sum is -.5 while each column sum is positive.
The problem with your "proof" is that it would not be a contradiction to have a matrix in which every row sum is positive and every column sum is negative.
Even if you do not care about showing absolute convergence for your end result, your "proof" absolutely requires it for there to be a contradiction when the sum is changed by having its terms reordered.
The series in your last post is absolutely convergent, as are each of the row sums in your matrix; thus they have a unique sum no matter the order of summation. The series including all terms in your matrix is, however, not absolutely convergent.
There is indeed an error. When an alternating sum is not absolutely convergent, there is no guarantee that a rearrangement of the terms will not yield a different sum. Thus nothing is proved when you assume the conjecture but get differing sums when you rearrange the terms.
For example...
This is just a standard exercise from enumerative combinatorics.
Label the urns 1 through k. How many ways can we place the M balls in the k urns so that each urn gets a ball? Line up the balls and urns in a straight line, with lower-labeled urns to the left of higher-labeled urns and with...
I'm a grad student at a pure math program, I only took the physics intro class as an undergrad, and I don't think many other people in my program have taken more physics than that, if any at all. The only time in a math class I felt physics classes would have helped was solving certain problems...
It does not necessarily carry out to infinity; induction only tells you your proposition holds for all natural numbers n.
Consider the proposition P(n) that the sum 1+2+...+n is finite. P(1) is true, and if P(n) holds, then 1+2+...+n is finite, so 1+2+...+n+(n+1) is also finite, so P(n+1)...
It is generally accepted that Fermat did not have a correct proof himself. He famously wrote his conjecture in the margin of a copy of a text he was reading, but wrote that even though he had a proof the margin was too small for him to write it there. He did have a proof for exponents equal to...
One easy way to get more comfortable with the solution to the Monty Hall problem is to imagine that there are 100 doors, only one of which has a car behind it. You pick a door, then the host opens 98 doors, none of which lead to a car. Any argument I've seen for the theory that no advantage is...
If it makes you feel any better about your chances, remember that being female will probably give your application a bit of a boost. (Full disclosure: I'm in grad school for math, not physics, but I'd imagine physics departments like to diversify their pool of grad students too.)