Intereting Posts

Solving $\lim_{n\to\infty}(n\int_0^{\pi/4}(\tan x)^ndx)$?
Proof by induction that if $n \in \mathbb N$ then it can be written as sum of different Fibonacci numbers
Expectation formula of an integrable random variable
The Group of order $p^3$
Symmetric Difference Identity
Linear Algebra, add these two matrices
Why $x^{p^n}-x+1$ is irreducible in ${\mathbb{F}_p}$ only when $n=1$ or $n=p=2$
Is there a domain in $\mathbb{R}^3$ with finite non-trivial $\pi_1$ but $H_1=0$?
Evaluate: $I=\int\limits_{0}^{\frac{\pi}{2}}\ln\frac{(1+\sin x)^{1+\cos x}}{1+\cos x}dx$
Find the expectation
Polar decomposition normal operator
Math story: Ten marriage candidates and 'greatest of all time'
Computing $\text{Gal}(\mathbb{Q}^{ab}/\mathbb{Q})$
Sum : $\sum \sin \left( \frac{(2\lfloor \sqrt{kn} \rfloor +1)\pi}{2n} \right)$.
closed-form expression for roots of a polynomial

Suppose we start at $(0.5,0.5)$ in an infinite unit square grid, and our goal is to traverse every square on the board.

At move $n$ one must take $a_n$ steps in one of the directions, north,south, east or west. And every square we walk over is marked as visited, we are not allowed to walk over a visited square twice.

Is there a sequence of directions, such that we can visit every square of the board exactly once if $a_n=n$?

- Maximum board position in 2048 game
- Prime Numbers and a Two-Player Game
- Name for a certain “product game”
- Extending Conway Games to $n$ players
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- Game: two pots with coins

Is there such a sequence if we are allowed to walk in diagonal directions aswell?

Is there a general algorithm to check, given $a_n$, if a path exists?

Is there a path in any of the above cases for $a_n=n^2$?

- Famous uses of the inclusion-exclusion principle?
- How to know if its permutation or combination?
- Combinatorial proof for $\sum_{k = 0}^n \binom {r+k} k = \binom {r + n + 1} n$
- Finding coefficient of a complicated binomial expression?
- Prove that $\sum\limits_{j=k}^n\,(-1)^{j-k}\,\binom{j}{k}\,\binom{2n-j}{j}\,2^{2(n-j)}=\binom{2n+1}{2k+1}$.
- What is the inclusion-exclusion principle for 4 sets?
- Comparing probabilities of drawing balls of certain color, with and without replacement
- Where are good resources to study combinatorics?
- Proof of $k {n\choose k} = n {n-1 \choose k-1}$ using direct proof
- Permutations with k inversions. Combinatorial proof.

If your $a_n$ are increasing, this is always impossible.

Suppose (by symmetry) that you start by going south. Sooner or later you will have to move north. However, after your first north move, you’ll have drawn an U shape of width $a_i$ on the grid, and there will be no way for you later to enter the *interior* of the U from the north and get back up again without having an $a_j$ available that is at most $a_i-2$.

This argument also *almost* shows that it is impossible with a merely *non-decreasing* sequence of $a_n$’s.

Things appear to be more murky if diagonal moves (like bishops in chess) are allowed.

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- Prove that an expression is zero for all sets of distinct $a_1, \dotsc, a_n\in\mathbb{C}$
- Let p be a prime. Consider the equation $\frac1x+\frac1y=\frac1p$. What are the solutions?
- Given continuity of measure, prove countable additivity to prove measure