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?

Six points are arranged as the vertices of a regular hexagon. A bug starts at one vertex. Each move, it randomly chooses one of the two adjacent vertices and walks there. After exactly 4 moves, what is the probability that the bug is back at its starting vertex?

Source: numberthon.com

Blah blah blah leading to endogenous variable monopolies profits becoming informational assets and then completeness of contracts under quota, no averse selection after negotiations ya da ya da dynamic programming and unforseen contingencies

Business cycle, product cycle, pricing cycle contained as barriers to market entry inter alia are quota equilibrium... It's difficult to explain but I worked on withstanding the monopoly until no more.

Want the math?

Twenty-five points are arranged in a 5×5 grid of equally spaced points (five rows and five columns). How many different squares (using four of these points as vertices) can be formed?

P.S. It's a 5x5 grid made up of 4x4 points/vertices.

Source: numberthon.com

Eight siblings – four brothers (Alan, Carl, Eric, George) and four sisters (Beth, Daniela, Fiona, Holly) – all have different ages. Within each group, the siblings happen to be arranged in alphabetical order of their names – which turns out to be the same as ascending order of age. Thus, among brothers, Alan is the youngest and George is the oldest, while among sisters, Beth is the youngest and Holly is the oldest.

The sum of the brothers' ages exceeds the sum of the sisters' ages by 10.

The following relationships between their ages hold:

• Beth's and Daniela's ages sum to Carl's age.
• Carl's and Beth's ages sum to Eric's age.
• Alan's and Carl's ages sum to George's age, and so do Daniela's and Fiona's ages.
• Alan's and Beth's ages sum to Fiona's age.
• Alan's and Fiona's ages sum to Holly's age.

Additionally, the product of Eric's and George's ages equals the product of Fiona's and Holly's ages.

Find the age of each sibling.

How many positive integers less than 100 are divisible by exactly one of 2 and 3?

Source: numberthon.com

A math competition committee of 3 people is to be chosen from a group of 5 teachers and 4 students. How many different committees can be formed if the committee must contain exactly 2 teachers and 1 student?

Source: numberthon.com

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.

Let a>0 be a fixed positive real number. Find all polynomials P with integer coefficients satisfying: gcd(P(m),P(n))>=gcd(m,n)^a for all positive integers m,n.

Let n and d be positive integers greater than 1. The numbers 1,2,...,n are written on a blackboard. In a move, we may pick two numbers on the board that differ by at least d, erase them both, and write their average instead. For a fixed d, let m be the smallest positive integer choice for n>1 such that it is possible to perform operations so that we end with exactly one number written on the board.

Show that: 3d - 2026 < m < 3d+2026.

How many positive integers less than 100 have an odd number of positive divisors?

Source: numberthon.com

How many positive integers less than 1000 are divisible by 6 but not by 9?

Source: numberthon.com

As I was falling asleep last night, I thought it was kinda cool that 56 = 7 * 8 works in base 10, specifically how it consists of four consecutive digits in order. Then I realized it actually happens again! 12 = 3 * 4

Is there any other base such that there are four consecutive digits A, B, C, D (in increasing order) such that AB = C * D? If so, are there any (besides base 10) where it happens twice? Why or why not?

I've been thinking about an interesting localization problem and I'm curious if there's a known solution.

Imagine a 100,000 × 100,000 grid. A single coordinate is chosen at random, but you don't know which one.

You may place as many fixed beacons as you want anywhere on or outside the grid. Each beacon tells you only the direction toward the hidden coordinate, rounded to the nearest 11.25° (so each beacon returns one of 32 compass directions). You get all beacon readings simultaneously.

Question: What's the minimum number of beacons needed to locate the target?

A few rules:

• Beacons are placed before the target is chosen.
• They never move.
• No distance information is provided—only the quantized direction.
• Your final guess is considered correct if it is within 1,000 grid units of the actual coordinate
• The beacon layout should also generalize to larger grids (i.e. not rely on the grid being exactly 100,000 × 100,000).

I'm interested in An actual beacon placement that achieves the minimum (or a proof that it can't). does anyone have ideas for constructing an optimal layout?

How many positive integers "n" less than 100 satisfy n² + n + 1 is divisible by 7?

Source: numberthon.com

Let k be a positive integer. Find the largest positive integer n such that the cells of an nxn grid can be filled with positive integers satisfying:

1) Each row and column contains the numbers 1,2,...,n in some order, and

2) The sum of numbers in any two kxk sub-squares is the same.

Note: A kxk sub-square is a contiguous kxk subgrid of the grid consisting of k^2 cells that are in k consecutive columns and k consecutive rows.

Daily math puzzle - https://quigly.app/

What it is: a daily card puzzle. 12 cards, four features each (shape, color, number, fill). Three cards make a trio when every feature is all-same or all-different. Clear the board in exactly four trios, the catch is that some perfectly valid trios are traps that strand the remaining cards. Same board for everyone, harder as the week goes on.

Can try all levels in the training grounds :)

In how many ways can 23 identical objects be shared among 5 children so that each child gets at least 2 and no child gets more than 6 objects?

Source: numberthon.com

Six variables 𝐴,𝐵,𝐶,𝐷,𝐸,𝐹 are distinct integers from 1 to 10 (inclusive).

They satisfy the following conditions:

1. B - D = 2
2. F + A = 11
3. A is between C and D (order of C and D not implied)
4. No two variables sum to 14
5. No two variables sum to 5
6. C − A = 1

Determine the value of the six variables.

This puzzle has exactly one solution, and it can be solved using logical deduction alone (no guessing or brute force required).

How would you solve this though a logical deduction sequence?

If you enjoy puzzles like this: https://sixfigurelogic.com/

A spider starts at the bottom-left corner of a 5 × 5 grid (5x5 vertices, 4x4 squares). It can only move up or right along grid lines. How many shortest paths to the top-right corner do not pass through the center point of the grid?

Source (where I got this specific variation from): numberthon.com

Ek accha sawal hai bhaiya

•A one-way road track is 20 km long and 8 km wide, divided into 4 equal lanes. There are 16 identical cars already on the track, moving at a constant speed of 10 km/h. Exactly 4 cars are present in each lane.

A new car enters the track from the starting point at a speed of 11 km/h. It chooses one of the four lanes uniformly at random and cannot change lanes thereafter.

Assume that the positions of the existing cars in each lane are independently and uniformly distributed along the length of the track, no two cars initially overlap, and overtaking is not allowed. A collision occurs if the new car catches up to at least one car in its lane before reaching the end of the track.

Find:

1.The probability P that the new car collides with at least one existing car.

2.The probability P' that the new car completes the journey without any collision.

a) P = (1/4 )⁴, P' =1- (1/4)⁴

b) P =( 1/11 )⁴, P' = 1-(1/11)⁴

c) P = (1/11)⁴ , P' = 1

d) P =1- (10/11)⁴ , P'=(10/11)⁴

Isko Maine khud banaya Hai Koi galti Ho To dekhna

How many three-digit palindromes are divisible by 9?

Source: numberthon.com

Q.1 Let ABCD be a trapezium in which AB k CD and AB = 3CD. Let E be the midpoint of the diagonal BD. If area ABCD = n×area CDE, what is the value of n?

Q.2 Let ABC be a triangle with AB = AC. Let D be a point on the segment BC such that BD = 48(1÷61) 61 and DC = 61. Let E be a point on AD such that CE is perpendicular to AD and DE = 11. Find AE.

Q.3 A 5-digit number (in base 10) has digits k, k + 1, k + 2, 3k, k + 3 in that order, from left to right. If this number is m2 for some natural number m, find the sum of the digits of m.

Q.4 Let ABC be a triangle with AB = 5, AC = 4, BC = 6. The internal angle bisector of C intersects the side AB at D. Points M and N are taken on sides BC and AC, respectively, such that DM k AC and DN k BC. If (MN)² = p/q where p and q are relatively prime positive integers then what is the sum of the digits of |p − q|?

Q.5 A group of women working together at the same rate can build a wall in 45 hours. When the work started, all the women did not start working together. They joined the work over a period of time, one by one, at equal intervals. Once at work, each one stayed till the work was complete. If the first woman worked 5 times as many hours as the last woman, for how many hours did the first woman work?

A large regular hexagon has an area of 120. Inside this hexagon, the midpoints of all six sides are connected (in order) to form a smaller, nested regular hexagon. What is the area of this smaller hexagon?

Source: numberthon.com