(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.
It was called:The waterist,so basically,there was a river,the river quantity was infinity,there was a group of people that wants to fill in the river called;waterist,there first fill was ½ of the river,the second was ½ of the first fill,mean ¼ of the river and so on,will waterist fill the dry river?
Can you reach the target number 257 using only the following four digits?
Given numbers are 2, 5, 6, and 8.
The only rule is you must use each of the four numbers exactly once.
(You may use +, −, ×, ÷, brackets, powers, and factorials.)
For years now, the evil mathematician wizard has been capturing and lining up groups of prisoners to let them guess the colors of the hats he put on them in exchange for their freedom. But since everybody nowadays already knows how to solve this problem, almost everybody escapes, prompting the wizard to come up with something more difficult. What if he used numbers instead of colors?
The next time he captures 1,000 prisoners, he lines them up in a row and gives everyone a hat with a positive integer written on it, subject to the following condition: The number of the first prisoner is at most 1, the number of the second one is at most 2, the number of the third one is at most 3, all the way to the 1,000th prisoner, whose number is at most 1,000.
Everything else is as usual:
What is the maximal number of prisoners that can be guaranteed to be freed?
Can you reach the target number 24 using only the following four digits?
Given numbers are 5, 5, 5, 1
Target is 24
The only rule is you must use each of the four numbers exactly once.
The Prime Anchor Theory This theory states that every composite number is held down by hidden "Prime Anchors." If you systematically pull out its prime factors, you can measure its "Gravitational Weight" and reduce any number back down to its absolute ground level: 1.
Can you reach the target number 37 using only the following four digits?
Given numbers are 1, 6, 6, 7.
Target is 37
The only rule is you must use each of the four numbers exactly once.
Let G be a finite graph and for every pair of vertices let P: Vert(G) \times Vert(G) \to [0, 1] be a matrix of transition probabilities for a random walk on G (so \sum_{x a neighbor of y} P(x, y) = 1) such that any vertex is reachable from any other vertex with nonzero probability. Assume as well that P(x, y) = P(y, x) for any x, y \in Vert(G), and that there is a group of symmetries \Gamma acting transitively on the vertices, such that P(\gamma x, \gamma y) = P(x, y) for all \gamma \in \Gamma.
Fix a vertex v_0 \in Vert(G), consider the two following games.
Game 1:
A token starts at vertex v_0 player 1 does a random step according to the probabilities P, then the next turn player 2 moves the same token again according to P, and so on. Each player gets a point for each vertex (not including v_0) that they visit first.
Game 2:
Each player has their own token both starting at v_0, player one moves her token according to P, then player 2 moves her token according to P, and so on. Scoring is the same.
Show the expected score difference of the two players is the same in both games.
Bonus: drop the condition that P(x, y) = P(y, x) and replace it with the condition \pi(x)P(x, y) = \pi(y) P(y, x) for all x, y, where \pi is the stationary distribution of the random walk P. Also drop the condition about the vertex transitive group of symmetries. Instead of fixing a vertex v_0, choose a vertex v_0 ~ \pi, and show that the same conclusion holds for the two games above, taking into account the random choice of v_0.
Possibilities: ████████████████████ 247
Eliminated: ░░░░░░░░░░░░░░░░░░ 0
Discussion Length: 6 hours
Progress: ?
This is a slightly modified problem from r/askmath (if you go searching for it, you’ll find my answer, so don’t spoil yourself).
Two players play a game as follows. There are n spots labeled 0 to n-1 in sequence around a circle, and both players start at 0. They alternate turns, starting with player 1, where a turn consists of flipping a coin to determine whether to move to the left or to the right one spot. Each non-zero spot awards 1 point to the first player to reach it, and the game ends when all spots have been visited. What is the expected (signed) point difference between player 1 and player 2?
EDIT: I should clarify that players move independently of each other, not as a group.
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?!<
Triangle ABC is an isosceles right triangle make of paper and D is the midpoint of leg AB. If the triangle is folded so C meets D, creased, and then unfolded, what is the ratio of the two segments the hypotenuse is split into by the crease?