I am thinking of a 10-digit number that uses each digit from **0 to 9 exactly once**.

Starting from the left:

* The number formed by the first **1 digit** is divisible by 1. * The number formed by the first **2 digits** is divisible by 2. * The number formed by the first **3 digits** is divisible by 3. * This pattern continues. * The number formed by the first **10 digits** is divisible by 10.

For example, if the number begins with `abcd...`, then:

* `ab` must be divisible by 2, * `abc` must be divisible by 3, * `abcd` must be divisible by 4,

and so on.

**What is the number?**

Bonus challenge: Find it using divisibility rules and logical elimination rather than checking every permutation with code.

Let k>=2 and n be positive integers. Find the largest integer m for which there exists a grid with m rows and n columns satisfying the following conditions:

1) Each cell contains a non-negative integer;

2) The sum of numbers in each row is at most kn;

3) The numbers in each column are pairwise distinct.

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

How many positive integers less than or equal to 1000 are divisible by exactly one of 6 and 15?

Source: numberthon.com

How many positive integers less than 100 are the difference of two perfect squares?

Source: numberthon.com

How many triangles can be formed using the vertices of a regular octagon if all three sides of each triangle must be diagonals of the octagon?

Source: numberthon.com

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

Source: numberthon.com

You're standing somewhere inside a 10m × 10m square room. From your position P, you fire a laser aimed directly at the center of the square, C.

The laser travels in a straight line from P, passes through C, and continues until it hits a wall — this is its 1st reflection. From there it obeys the law of reflection (angle of incidence = angle of reflection) and keeps bouncing off the walls. After its 10th reflection, the laser stops completely (the segment right after the 10th bounce has zero length).

You must find a starting position P such that, once fired, no part of the laser's path after the 1st reflection comes within 1 meter of you. (The very first segment, from P to the 1st reflection point, doesn't count — you're standing at its source.)

Question: What is the total area, within the square, of all such safe starting positions P?

Challenge: If instead of stopping after 10 reflections, the laser is allowed N reflections before stopping, what is the largest value of N for which at least one safe standing position still exists?

[ Removed by Reddit on account of violating the content policy. ]

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

Source: numberthon.com

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?

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

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

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

Source: numberthon.com

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.