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.

During one of your long journeys, you come across a desolate mountaintop in the middle of nowhere. To your surprise, you are met with a crowd of gods arguing with each other. Upon seeing you, they stop their debate to tell you that they will grant you a wish if you can solve their riddle.​

There are 100 gods in total. When asked a question, 90 of them will always answer truthfully, while the remaining 10 will answer each question randomly. Initially, you don’t know which god falls into which of the two categories, but each god knows it for every other god.

Your goal is to identify at least one truth-telling god using at most 20 yes-or-no questions. Each question can be directed at only one god at a time, though multiple gods can be asked the same question one after another, with each asking counting towards the limit.

Can you come up with a strategy that is guaranteed to work?

Even though I coded this interactive riddle myself, this is the only one from this library which I still haven't solved. The other puzzles were much easier to figure out.

You can try it yourself here:

https://www.problems.cc/p/YHqypk?from=library&slug=mathematics-in-chess-maximum-non-attacking-pieces

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.

You are given an n-sided die (taking values uniformly in 0,…,n-1), and your goal is to simulate a k-sided die with a sequence of dice rolls. Specifically, you must choose a procedure which

• decides at each step whether to roll again or to terminate and decide a value based on all previous rolls
• is deterministic in the sequence of rolls
• terminates almost surely
• produces a value uniformly distributed in 0,…,k-1

We will call such a procedure an (n,k)-procedure and the number of rolls before it terminates the decision time (in general, the decision time depends on the sequence of rolls; the decision time is infinite if it does not terminate). We will assume n,k≥2.

1. For fixed n,k, determine the minimum expected decision time among all (n,k)-procedures. Give your answer as a finite sum depending on n,k.
2. Determine for which n,k, there exists an (n,k)-procedure that always terminates.

(this is a follow up question from this question inspired by u/blungbat)

the image shows 3 rectangular grids on torus, i.e. (Z/nZ)^2 .

each torus is partitioned into two regions. each region is connected and does not contain 2x2 subregion.

unfortunately the second and third grid is partially destroyed. their size is 7x7 and 8x8 in an unknown order.

how many blue tiles are there in each torus?

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.

Can you work this out? This actually happened in my kitchen when I was a teenager.

When I went to bed, at 11pm, all three clocks in the kitchen were working and said the correct time: 11pm.

When I woke and went into the kitchen for breakfast, all three clocks were working, yet one said 5am, one said 6am and the third said 7am. No one had touched them.

Why? How? What happened and when? And what was the correct time when I went down for breakfast?

SOLUTION:

(Congrats u/RealHuman_NotAShrew)

There was a power cut for one hour at 1am. The battery powered clock on the wall continued unaffected. The analogue clock on the oven froze for the hour then resumed at 2am, one hour behind. The digital clock on the microwave turned off at 1pm then reset at 2am to 00:00. 2 hours behind. At 7am, the wall clock said 7am, the analogue clock on the oven said 6am and the microwave clock said 5am.

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?

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 $S$?

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.)

Rules are fully stated below, there is no hidden rule to guess.

A honeycomb hides one secret 6-letter word. Every cell holds either a letter or a number. A number is a Pollen count: it equals how many of that cell's neighbours hold a letter FROM THE SECRET WORD. Five of the letters on the board are impostors; they sit there like any other letter and count toward nothing. Adjacency: two cells are neighbours iff they are in the same column two rows apart, or in adjacent columns one row apart (so an interior cell touches six, edge cells fewer).

___ ___/ N ___ ___/ 2 ___/ C ___ / 1 ___/ E ___/ M \ ___/ R ___/ 2 ___/ / U ___/ 3 ___/ 1 \ ___/ A ___/ W ___/ / T ___/ 2 ___/ 0 \ ___/ D ___/ 1 ___/ ___/ K ___/ ___/

Only the numbers separate the real letters from the impostors, and every number earns its place: the eleven letters on this board also spell MARKED, RACKET and UNMAKE, and it takes the whole grid to rule those out. The solution is unique. What is the secret word?

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

The sum of the digits of a number N is 23. The remainder when N is divided by 11 is 7. What is the remainder when N is divided by 33?

7

29

16

13

Ans is 29 if somebody has the easiest way of doing this question please let me know

English version is here : https://www.fichier-pdf.fr/2026/10/02/the-magic-square20260925/

This rather surprising 4x4 magic square features 13 cells that revolve around three constants: 11, 13, and 15. By extension, 24 (11 + 13) and 28 (15 + 13) also appear.

This

I do not think I have discovered everything in this magic square, which is why I am asking for your help.

This is a question from OpenQuant site. It was rated 8/10, but i believe, if you are even a bit aware of that specific topic, you'd be able to solve it.

A baby is learning to walk with the assistance of its living room couch. The baby starts at the couch and at each given time the baby will make a decision. It will either take a brave step forward, stand in place not knowing what to do, or fearfully take a step back towards the couch with probabilities 0.2, 0.5, and 0.3 respectively. The baby will never go behind the couch (so when at the couch the baby has probability 0.2of moving forward and probability 0.8 of staying at the couch).

If you were to observe this baby for an extremely long amount of time, what proportion of the time would the baby be at the couch?

Hint: that specific topic is Markov Chains

Clarifications:
- A domain is a non-empty connected open set
- A chord is a (possibly degenerate) line segment with both endpoints on a curve

EDIT: assume K is bounded

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?

Imagine a universe where spatial positions are represented as 2x2 real matrices X in M_2(R) rather than standard spatial vectors.

1. How many total independent 2D planes of rotation exist in this space?

2. What is the maximum number of mutually orthogonal rotation planes a planet can orbit across simultaneously?

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?

Oki, so there's a guy who has 17 camels, he passes away and writes in his will that the eldest son will get 1/2 of the camels, the second son will get 1/3, and the youngest will get 1/9. There are only 3 sons who will inherit, and no other family members whatsoever. The problem now is that they all want whole camels and do not want to sacrifice and distribute any camel. How would they solve this distribution issue?

Answer: They borrow another camel from somewhere so now the total is 18. This can easily be distributed in the fractions needed. 1/2 = 18/2 = 9 1/3 = 18/3 = 6 1/9 = 18/9 = 2

Adding them all now makes 9 + 6 + 2 = 17 So they return the 18th camel that they borrowed and now all of them have the fractions their father left for them.

I cannot wrap my head around why dividing 18 and then adding them all makes 17.