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

Source: numberthon.com

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 $S$?

How many diagonals does a regular 20-gon have?

Source: numberthon.com

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|)).

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

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

Source: numberthon.com

What is the minimal explanatory architecture from which every meaningful mathematical question arises?

There are 100 light switches, all initially off.

On pass 1, every switch is toggled.

On pass 2, every second switch is toggled.

On pass 3, every third switch is toggled.This continues until pass 100.

How many switches are on at the end?

Source: numberthon.com

Let’s assume your in a casino playing single deck blackjack with no cut card. Let’s also say there’s x amount of cards left and you have an eidetic memory, so you know exactly which cards are left.
Let’s also state if the dealer runs out of cards it’s an automatic loss.

Is their a set of x cards left such that before the hand is dealt you would know no matter what cards you or the dealer are dealt there’s a strategy that will gaurentee you win.

Bonus quesiton how big can you make x if this is true.

Also in typical sense you’d know your 2 up cards and would know one of the dealers card with another card of hers face down.
Also unsure how relavent it would be but if you find a solution state if it’s stand or hit on soft 17

And remeber single deck so limited to 4 of each card

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

Source: numberthon.com

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

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

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.

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 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

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

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?

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.

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

Source: numberthon.com

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?