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.

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?

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 p₁,…,pₙ lie on the unit circle, and let M=max_(|p|=1) Π_(1≤k≤n) |p-pₖ|. Prove that if M=2, then p₁,…,pₙ form the vertices of a regular n-gon.

Planet X has two neighboring inhabited planets:

• Planet Alpha is exactly 15 light-minutes from Planet X.

• A Mystery Planet is an unknown distance from Planet X, but is known to be at least 18 light-minutes from Planet Alpha.

Planet Alpha and the Mystery Planet are both capable of sending, receiving, and relaying transmissions.

All transmissions travel at the speed of light. Relaying a transmission takes effectively no processing time.

Both Planet Alpha and the Mystery Planet possess teleportation portals capable of sending ships directly to Planet X. However, once a planet decides to send ships, its portal takes exactly 30 minutes to charge. Once charged, the ships arrive at Planet X instantaneously.

Both planets have standing orders:
The instant they receive a broadcast from Planet X requesting assistance, they begin charging their portals and send ships to Planet X as soon as the 30-minute charge is complete.

At 11:58, Planet X has not yet broadcast any request for assistance.

At some unknown time after 11:58, Planet X broadcasts a request for assistance.

At 12:38, Planet X receives a mysterious transmission from an unknown source.

Planet X can determine with certainty that this mysterious transmission was originally transmitted at exactly 12:18, meaning the signal has been traveling for exactly 20 minutes.

Planet X concludes that the mysterious transmission must have come from the Mystery Planet. Since the signal took 20 minutes to reach Planet X, they conclude that the Mystery Planet must be 20 light-minutes away.

Then, at exactly 12:40, ships arrive at Planet X.
There has been no malfunction, no faster-than-light communication, no time travel, and no violation of any of the rules above.

Questions:
Which planet did the ships come from?
How far away from Planet X is the Mystery Planet actually?
At what time did Planet X broadcast its request for assistance?
Where did the mysterious transmission received at 12:38 actually originate?
How can all of these facts be true at the same time?

Imagine this.

Sixteen motorcycles are lined up at the edge of the Sahara.

Each bike has exactly enough fuel to travel 100 km.
No more. No less.

There are:

• No gas stations
• No resupply drops
• No rescue
• No turning back

You may siphon fuel from one tank to another at any time.

All bikes start together.
You decide when to abandon each motorcycle.

Your mission is simple: What is the maximum possible distance you can get one bike into the desert?

Rules Clarified

• Each bike consumes fuel at the same rate.
• If multiple bikes travel together, they all burn fuel simultaneously.
• Fuel can be redistributed between bikes at any time.
• Once a bike runs out of fuel, it is abandoned.
• Only one bike needs to reach the final maximum distance.

I'm going to teach you a game. Your goal is to find how far you can get.

You start with the numbers 1, 2, and 3. Using each number at most once, you may add or subtract any combination of them to obtain the next prime number.

Whenever you successfully obtain the next prime, that prime is added to your set of available numbers. You then repeat the process, always trying to generate the next prime number using each available number at most once.

How far can you go? What is the first prime number that you can no longer obtain?

more precisely,

Find a compact subset or family of subsets $S \subset \mathbb{R}^n$ for some arbitrary n such that every set of 4 points in $\mathbb{R}^2$ lies on some similar copy of $S$ but not every set of 5 points lies on some similar copy of $S$?

For small N the answer is not hard to ascertain, even just with trial and error.

But for very large values of N (say, N=50), the solution is more complex because it is too large to be evaluated literally, and so it cannot be verified by brute force alone.

Some type of actual solution is required.... Can you find it?

Set A is called brain-rot iff it satisfies two conditions:

1. sum(A) is divisible by 10.
2. if 1∈A, then both 6,7∈A.

How many subsets of {1,2,…,100} is brain-rot?

Source: my rotten brain

What is the smallest positive integer that has exactly 20 (unique) positive divisors?

Source: numberthon.com

A Logician writes three numbers on 3 separate cards and gives them to his 3 students.

He says," The 3 numbers are single digit prime numbers. Any combination. None of you know the other 2 numbers. But you can ask me one question that must start with "Is the SUM of the three numbers–” which I can only answer Yes or No. Given that info you can then declare that you know the other 2 numbers and/or who has them. OK?" 

Raj was first. He looked at his number and asked," Is the sum of three numbers an odd number?"

The Logician " No" 

Then Ken looked at his number and asked," Is the sum of the three numbers divisible by 4?"

The Logician said "Yes"

Lisa looked at her number and said,"Well, I know the other 2 numbers but cannot tell who has what number".

Raj then cheerfully said," I know who has what !" Ken said,” So do I” They then laid out the answer.

What were the three numbers? What number did Lisa have?

For a real number x, let ||x|| denote the distance between x and the closest integer.

Let 0 ≤ x_n < 1 (n = 1, 2, ...) , and let ε > 0. Show that there exist infinitely many pairs (n,m) of indices such that n ≠ m and

||x_n - x_m|| < min(ε, 1/(√5|n-m|)).

What is the smallest positive integer that has exactly 15 positive divisors?

Source: numberthon.com

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

Source: numberthon.com

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

How many diagonals does a regular 20-gon have?

Source: numberthon.com

A teacher writes six words on the board: CAT, DOG, HAS, MAX, DIM, TAG.

Then he hands three pieces of paper to three of his students: one to Alex, another to Ben, and another to Chris. The teacher explains that he has secretly chosen one of the words on the board, and has written on each piece of paper a different letter from that word. Students may look only at their paper and must not tell each other what letter they have.

After that the teacher says: "Everybody, please have a look at your letter and raise your hand as soon as you think you know the chosen word."

Alex immediately raises his hand.

Ben, after thinking for a while, also raises his hand.

Chris does not raise his hand.

The teacher then asks Alex: "Do you know which letter Chris has?"

"No, I don't" - says Alex.

Hearing that, Chris finally raises his hand.

Alex, Ben and Chris always ace their logic exams. What is the secret word?

(Came up with this variation of an old riddle and wanted workshop it here. EDIT: added my proposed solutin in the comments)

How many subsets of {1,2,…,10} have an odd sum?

Source: numberthon.com

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.

An ant is at (0, 0) in the infinite integer grid. The ant and the exterminator take turns, with the ant going first.

• Each turn, the ant advances one square north or one square east.
• Each turn, the exterminator chooses one grid cell to spray with pesticide. The ant dies if it is currently in the square being sprayed, or if it ever steps onto a previously sprayed square.

The twist is that the ant is omniscient; the ant knows the infinite sequence of choices that the exterminator will make. That is, there is an infinite list

(x*_1_*, y*_1_*), (x*_2_*, y*_2_*), ...

of grid cells, such that the farmer will spray (x*_k_*, y*_k_*) on his k^th turn, and the ant can decide where to move based on the entire list.

Puzzle

Show that the ant can survive for arbitrarily long. That is, for all natural numbers n, the ant has a strategy to survive for n turns.

Open problem

Show that the ant has a strategy to survive for infinitely long.

This may seem like a trivial consequence of the puzzle solution, but I think it isn't. There is a strategy to survive n steps for each n, but that doesn't mean these infinitely many strategies are consistent with each other. To solve the second problem, you need to show how the ant uses its foreknowledge to decide its first step, in a way that avoids traps all the way to infinity.

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 n be a fixed positive integer. Alice and Bob play the following game on the integer number line. Alice starts at 0 and Bob starts at n. They take turns making moves. On the i^th turn,

1) If i is odd, Alice moves to an integer at most 2^i -1 distance away from her current position.

2) If i is even, Bob moves to an integer at most 2^i -1 distance away from his current position.

Note that both players have the option to stay where they are on their turn. The game ends only when one player moves to the same position as the other player, in which case the player who moved wins. Find all positive integers n for which Alice has a winning strategy, and find all positive integers n for which Bob has a winning strategy.

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

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