(easy) show that a unit square can fit inside the region [0,x]^3 where x = >!2 sqrt2 / 3 ≈ 0.94281!< .
(bonus) show true or false: x is the minimum. i strongly believe this is true but i have no proof of it.
This is a slightly modified problem from a recently problem , where i misinterpret as two players moving a stone around a C_n graph. To spell it out:
Two players play a game as follows. There are n nodes around a circle, a stone is placed at one of the node. Player alternate turns, moving the stone to one of the two adjacent nodes with equal probability. Each non initial node awards 1 point to the first player to reach it, the game ends when all nodes have been visited. What is the expected (signed) point difference between the players?
alternatively, prove that the expected difference is >!if n is even then 1 else 1-1/n. which is surprising because this answer is same as the original problem. maybe there is a connection that transform the two variant?!<
Set A is called brain-rot iff it satisfies two conditions:
How many subsets of {1,2,…,100} is brain-rot?
Source: my rotten brain
given that the sum of 1/|u|^4 over all u ∈ Z^2 \ {(0,0)} is equal to (2/3) G π^2 .
find the sum of 1/|u|^4 over all u ∈ K , where K is the set of these 8 points tiling over Z^2 by translating 5 units in four coordinate-axis directions.
alternatively, prove that the sum is equal to >!(32/625) G π^2!< .
^(note: i discovered a trick while trying to solve a related problem posted here awhile ago. while it is cute, it failed to work, so i repurposed it and design a new problem around it.)
a pawn is placed in a cell of an infinite hexagonal board. a mitosis move consists of removing a pawn and placing two pawns on an empty cell adjacent to the removed pawn. pawns cannot share a cell, so a pawn cannot perform mitosis if there are less than two empty adjacent cell.
the goal is to clear a region D(n) such that no pawn is inside D(n), where D(n) is the set of cells whose shortest distance to the initial cell is <= n.
for example, D(1) is clearable with 3 mitosis moves.
(a) prove that D(6) is not clearable.
(b) prove that D(4) is clearable.
(c) is D(5) clearable? i actually don't know the answer to this.
In a m by n chess board, place m kings on the leftmost column. Count the number of ways to move all kings to the rightmost column, such that each tile is visited at most once.
Note: a king moves one tile to one of the 8 adjacent tile.
construct a n by n table with these conditions:
for odd n, this is relatively easy.
for even n, i cant figure out a way. i suspect there is no solution but i cannot prove it.
unrelated note: this was inspired by IRL problem that i had to solved when creating a duty time table. if you know a working solution for n=8 please show it.