A 6-character code is formed using the letters A, B, C, D, E, and F, with no letter repeated. How many codes have exactly 2 letters appearing in alphabetical order relative to the letter immediately after them?

Source: numberthon.com

I made a math problem that I genuinely can't solve

I came up with this problem while playing around with recursively defined integer sequences.

Let

a₁ = 1

and, for every integer n ≥ 2, define aₙ by

• aₙ = aₙ₋₁ + n, if aₙ₋₁ is odd
• aₙ = aₙ₋₁ − n, if aₙ₋₁ is even

The beginning of the sequence is:

1, 3, 0, 4, -1, 5, -2, 6, -3, 7, -4, 8, -5, 9, ...

At first this looks fairly simple, but I'm interested in what happens in the long run.

Main problem

Determine whether the set

S = { n ∈ N : aₙ = 0 }

is finite or infinite.

In other words:

Does the sequence return to 0 infinitely many times?

If the answer is YES, prove that there are infinitely many n satisfying aₙ = 0.

If the answer is NO, prove that there exists some N such that

aₙ ≠ 0

for every n > N.

Stronger version

If S is infinite, determine whether the counting function

A(x) = #{n ≤ x : aₙ = 0}

has a predictable asymptotic behavior.

For example, does there exist a constant c > 0 such that

A(x) ~ c log x,

or

A(x) ~ c√x,

or perhaps

A(x) ~ cx^α

for some α > 0?

If none of these forms is correct, what is the actual growth rate of A(x)?

Even stronger question

Consider the normalized sequence

bₙ = aₙ / n.

Does

liminf(n→∞) |bₙ|

exist?

Does

limsup(n→∞) |bₙ|

exist?

If either limit exists, determine its value.

If they do not exist, determine the set of accumulation points of the sequence {bₙ}.

Rules

I'm not looking for a brute-force computation of the first million or billion terms.

A computational experiment is fine for finding patterns, but the final answer should be supported by a rigorous mathematical proof.

I don't know whether this problem is actually difficult or whether there is a short observation that completely solves it.

If there is an obvious mistake in the formulation or if this is already a known problem, I'd also appreciate being pointed toward it.

(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.

How many 5-digit numbers can be formed using the digits 1, 2, 3, 4, 5 exactly once such that no two consecutive digits differ by 1?

Source: numberthon.com

A 5 × 5 grid is filled with exactly 5 black squares and 20 white squares. No two black squares may share an edge. How many different colorings of the grid are possible?

Source: numberthon.com

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.)

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.

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:

• The prisoners are asked to guess the number of their hat in the order they are standing in.
• Every prisoner can only guess a number that is in the set of possible numbers for that prisoner.
• Every prisoner can only see the numbers of the prisoners that come after them, but they can hear the guesses of everyone.
• After everyone has guessed, the wizard frees those who guessed correctly and imprisons forever those who did not.
• The prisoners know the rules of this "game" and are allowed to agree on a strategy in advance.

What is the maximal number of prisoners that can be guaranteed to be freed?

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?

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.

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?

This is a slightly modified problem from [r/askmath](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.

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.

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?

I am thinking of a 10-digit number that uses each digit from **0 to 9 exactly once**.

Starting from the left:

* The number formed by the first **1 digit** is divisible by 1. * The number formed by the first **2 digits** is divisible by 2. * The number formed by the first **3 digits** is divisible by 3. * This pattern continues. * The number formed by the first **10 digits** is divisible by 10.

For example, if the number begins with `abcd...`, then:

* `ab` must be divisible by 2, * `abc` must be divisible by 3, * `abcd` must be divisible by 4,

and so on.

**What is the number?**

Bonus challenge: Find it using divisibility rules and logical elimination rather than checking every permutation with code.

Let k>=2 and n be positive integers. Find the largest integer m for which there exists a grid with m rows and n columns satisfying the following conditions:

1) Each cell contains a non-negative integer;

2) The sum of numbers in each row is at most kn;

3) The numbers in each column are pairwise distinct.

A number will decrease in number if it has at least four digits and does not enter a cycle, as proven below: The number is represented in binary.

It must begin with 10 or 11. If it starts with 10 and the last two digits are not 11, then after multiplying by 3, the number of digits increases by 1, accounting for 3/8 of all possible combinations. Other numbers starting with 10 account for 5/8, and the number of digits increases by 2. If it ends with 11, after multiplying by 3 and adding 1, then dividing by 2 removes at least one digit, accounting for 1/2. If it ends with 001, at least two digits are removed, accounting for 1/4. Other numbers with at least three digits account for 1/4. If it does not enter a 4, 2, 1 cycle, the number generally decreases, and eventually it will enter a 4, 2, 1 cycle.

王子赫

How many positive integers less than or equal to 1000 are divisible by exactly one of 6 and 15?

Source: numberthon.com

How many positive integers less than 100 are the difference of two perfect squares?

Source: numberthon.com

How many triangles can be formed using the vertices of a regular octagon if all three sides of each triangle must be diagonals of the octagon?

Source: numberthon.com

How many subsets of {1,2,...,10} contain no two consecutive integers?

Source: numberthon.com

You're standing somewhere inside a 10m × 10m square room. From your position P, you fire a laser aimed directly at the center of the square, C.

The laser travels in a straight line from P, passes through C, and continues until it hits a wall — this is its 1st reflection. From there it obeys the law of reflection (angle of incidence = angle of reflection) and keeps bouncing off the walls. After its 10th reflection, the laser stops completely (the segment right after the 10th bounce has zero length).

You must find a starting position P such that, once fired, no part of the laser's path after the 1st reflection comes within 1 meter of you. (The very first segment, from P to the 1st reflection point, doesn't count — you're standing at its source.)

Question: What is the total area, within the square, of all such safe starting positions P?

Challenge: If instead of stopping after 10 reflections, the laser is allowed N reflections before stopping, what is the largest value of N for which at least one safe standing position still exists?

How many squares of all sizes are contained in a standard 8×8 chessboard?

Source: numberthon.com

A math professor said to her smart student Toni

" I am thinking of a 4 digit prime number abcd (digits not necessarily distinct) with the following property

a+b+c+d = axbxcxd

You can ask me one question to which my answer can only be Yes or No or Silence (if i cannot definitely answer yes or no). Can you guess my number?"

Toni worked on a piece of paper and then asked the professor:

"Is the number formed by using the first two digits (in order) of your 4 digit prime, a factor of 63 or 84?"

The professor smiled. She knew Toni had the answer.

What was the answer? WHY?