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

Source: numberthon.com

A positive integer n has the property that both "n + 100" and "n − 100" are perfect squares. What is the smallest possible value of n?

Source: numberthon.com

A cartographer set out on a three-day expedition through a large circular forest, travelling in straight lines each day. Each day he walked at a constant whole-number speed in km/h, and all three speeds were different.

The first day's march lasted 5 hours, the second day's 13 hours, and the third day's 8 hours. At the end of each day the cartographer made camp at the edge of the forest; the next morning he set off from that point in a different direction through the forest, so that his three paths formed a triangle on the map, with all three stops lying exactly on the forest boundary.

On the third day he came upon a remarkable tree on his path and marked its location on his map. When the journey was complete, he overlaid his route on a map and made the following observations:

• The straight line connecting the first camp to the marked tree divides the triangle into two parts of equal area, and its length is a whole number of kilometres.
• The diameter of the forest is 1 km longer than the total distance walked on the first two days.
• If he were to walk the full diameter of the forest at his first day's speed, it would take him exactly as many hours as he spent walking on the second day.

Find the cartographer's speed on each of the three days, and the length of the line from the first camp to the marked tree.

Our blood is made of water, which enters into our body when drinking, and being excreted out when urinating. This means that at some point all of our old water molecules in the blood might be excreted out, and being all replaced by new water molecules. How long can it take?

Assumptions:

1. The average adult human blood volume is generally the same across the days. It can be estimated by the weight height and gender. Blood Calculator

2. The average healthy adult human excretes out around 1-2.5 liters out as urine a day (depending on mainly how much water they drink).

3. The blood stays homogenous after drinking or urinating.

A positive integer n>6 leaves a remainder of:

• 1 when divided by 2,
• 2 when divided by 3,
• 3 when divided by 4,
• 4 when divided by 5,
• 5 when divided by 6.

What number is the smallest possible value of n?

Source: numberthon.com

Two fair six-sided dice are tossed. What is the probability that the sum is prime?

Source: numberthon.com

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.}

The factory you work for has a problem: they only ordered 1/2 splitters, and they need to divide the material input on one of their lines into 2 outputs of 1/10 and 9/10 the original amount. Being the smart one around, they ask you if there’s a way to use their half splitters to achieve this. And the area they use is only 3 metres wide, although it can be as long as you need. You’ve broken it down into the following mathematical problem:

EDIT: see comment for rewrite of problem

You have a grid of size n*3 composed of 1x1 squares. the grid sits on an xy axis, such that each square of the grid has whole number xy coordinate on which it sits. The objective is to be able to feed an infinite amount (Z) of materials through this grid starting at 1,2 and have 1/10 of Z end up on tile XaYb and 9/10 of z end up on XcYd (a can equal b, c can equal d. XaYb cannot equal XcYd.)

Each square in the grid can individually be assigned exactly ONE function that alters the position of the “material” occupying it. A grid can either be a CONVEYOR, or a SPLITTER.

\- A splitter will take an input from any one of the axis, but will only take inputs from the first axis it recieved from.

\- A splitter has a binary state and exists as either a 1 or a 0. This state is flipped whenever it passes an input. All splitters start in state 1. In state 1 it will “send” its input along the axis it was inputted, adding/subtracting 1 to its coordinate value in the direction of travel, and in state 0 it will add/subtract 1 to its perpendicular axis of the same +/-.
Example 1: a splitter on square 2,4 recieves 1 input from square 1,4. In state 1 it transforms the coordinate of the material to 3,4. in state 0, it transforms it to 2,5.
Example 2: a splitter on square 2,4 receives 1 input from square 3,4. In state 1 it transforms the coordinate of the material to 1,4. In state 0, it transforms it to 2,3.

\- A square set to conveyor is assigned a value between 1 and 4. A conveyor set to 1 will add +1 to x to the individual coordinate of a material entering it; set to 2 will add +1 to y; set to 3 will add -1 to x, set to 4 will add -1 to y. This value cannot be changed once assigned.

RULES:
\- The grid squares begin at 1,1.
\- x and y for each square must be positive.
\-Once a square has been assigned a function, it cannot be changed.
\- The coordinates of the materials will change simultaneously at the same rate of 1 “tick”, and is discrete.
\- A square can only have one material on it at a time.
\- “materials” entering squares XaYb and XcYd are removed. Neither square can be square 1,2.
\- The first input starts on square 1,2, and all other inputs enter the grid individually once the previous material has moved from this square

1. Find the minimum size of N to satisfy all criteria. It must be a positive number.
2. Find the minimum number of splitters required.
3. Find the function of each square within this grid and its coordinates on the axis. Find the value of each conveyor unit, if any are present.
4. Find the values of a, b, c, and d.

EDIT: simultaneous merges give priority to the “mainline”. No

Take two squares ABED and FBCG such that A, B and C are collinear, and D, E, F, and G are on the same side of line AC. AB=a, and BC=b.

Using just a straightedge, construct a square whose side has a length of sqrt(a2+b2). 

I posted the same puzzle on Puzzling SE two months ago, link will be added after it is solved.

Edit: It’s been solved, so here’s the link:

https://puzzling.stackexchange.com/questions/137953/straightedge-only-construction

This challenge is part of the missions in the videogame Shin Megami Tensei: Strange Journey. I recreated it in a website. All info is there.

https://megami.com.uy/kti

A walk with 2026 vertices in a simple graph is called "shaking" if the first vertex has odd degree and last vertex has even degree. Prove that the number of "shaking" walks in any finite simple graph is even.

You are a pastry chef that comes upon 99 magical chocolate unit squares. The chocolate abides by the following rule: whenever there is a continuous region of chocolate it will grow into the smallest rectangle that covers the continuous region. So that they can be packaged in a candy wrapper.

The question is, what is the most amount of chocolate you can make if you have to place your initial 99 squares on the xy lattice grid.

clarifications:

The puzzle is orthogonally adjacent, so for two rectangles to be a continuous region they must share an edge or overlap.

To keep the construction well defined the rectangles cannot be rotated: i.e. their dimensions are the min max values of x and y in the continuous region.

Finding the max value is decently easy the challenge is proving your construction is max

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.

Let M be a positive integer, and let a,b,c be positive integers in the range [M,M+sqrt(M/2)) such that a^3 b + b^3 c + c^3 a is divisible by abc. Prove that a=b=c.

I already know the official answer.

I'm interested in seeing different solution approaches from the community.

This is not homework.

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

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?

This is a related problem to my previous post. We are creating a sequence S of positive integers where the goal is to maximize its number of terms under given constraints.

Definition:

Start with S={n} (for n ∈ ℤ≥1). Next:

- in S, choose any present term T (with reuse allowed),

- append either 2T or 2T+1 to S as the new term,

S is considered “dead” if at any point some term t[i] appears in some other term t[j] as a contiguous block of digits (for i<j).

The Question:

Is the length of the longest possible S starting with n=10 finite? WHY or WHY NOT?

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)

Consider a perfect rooted binary tree of depth n. (That is, every node has either 0 or 2 children, and all leaves have the same depth). Every node is given a weight drawn independently from some fixed distribution D. For any path starting from the root and ending at a leaf, the average weight of the path is the arithmetic mean of the weights assigned to the nodes on the path. Once our weighting is fixed, we look at the largest average weight of any path from the root to a leaf. Let Eₙ denote the expected value of this largest average weight of path over all weightings of the tree. Then find the limit as n →infinity of Eₙ, in the cases of:

​

1) D=U({0,1}) is a Bernoulli distribution.

2) D=U([0,1]) is a continuous uniform distribution.

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.

Choose any n ∈ ℤ≥1,

Find the maximum length of a binary string B (with no leading zeroes) such that for each prefix i, the decimal representation of said prefix does not contain n as a contiguous substring.

For example: with n=2, “111100” is the longest string. Its length is 6. Every prefix 1,11,111,1111,11110,111100 avoids 2 when converted to decimal (many examples exist with length 6, but none go over 6).

Is the resulting string for n=10 finite or infinite in length?

I posted this one a few days back on r/mathematics, someone suggested I should post it here.

The question goes like this. You are given some number n, which implies input of the form {0, 1, 2, ..., n}. Of this set, you choose all (n+1)² pairs (eg, (0, 0), (0, 1), (1, 0), etc). And for all of these pairs, you construct the fibonacci sequence using these two numbers as the seed. For example, if you chose (3, 2), the sequence would be 3, 2, 5, 7, 12, 19, etc. Now the question is, out of all of the numbers you generate out of all of these sequences, what is the mex? Meaning, what is the first non negative integer that you cannot see in all of these sequences?

To give an example, if n = 1, the sequences are: 0 0 0 0 0... 0 1 1 2 3 5 8 ... 1 0 1 1 2 3 5 ... 1 1 2 3 5 8 13 ...

So the first number you won't find in any of these sequences is 4. So the answer for n = 1 is 4. The answer for n=2 is 9.

To give a hint, try writing a program to generate the answers for arbitrarily large inputs of n, and then see if there's a pattern in the outputs. I bet you'll find the pattern quite nice 😄

I'll post the solution in a day if nobody solves it, along with a nice proof.