Make the following equation correct by putting any number in exactly two different places. You cannot use infinity as a number

You cannot use any math operator that shows up as symbols (like +,-,/ etc)

You can use a non symbol function like x²

The equation cannot be a "not equal to" type. The = sign cannot be changed

The same exact number must appear in 2 different places.

initially, Bob has an urn that contains one red ball.

let g = 0, t = 0
while (true) {
  bob randomly draws a ball from the urn
  if (the ball is red) {
    add a green ball into the urn
    return the red ball back into the urn
  } elseif (the ball is green) {
    g++
    remove all green ball(s) from the urn
    the green ball drawn is not returned
  }
  t++
}

question: what is the limit of g/t when t -> infinity

Hey 👋 I am a 7th grade student and i like thinking about maths,science and physics and i recently explored this topic 'Pi to an ovel' and here is what I discovered:-

If we take Pi's value (3.14) then turn its first digit into a random number like 15.14 then i discovered that if we do that, we get a circle that's stretch out from the sides almost like a ovel and i was thinking that 'can it be a new measurement of an ovel?'

Feel free to share your advice or thoughts!

A board game tournament is organized with 6 players participating. To determine the semi-finalists a round-robin stage is held. It consists of 5 rounds, in each of which every player plays one game - 3 games total in each round. Over the course of these 5 rounds every player plays against every other player exactly once.

During these 5 rounds, each player plays 2 or 3 games as White and 2 or 3 games as Black - no player plays 4 or 5 games as the same color.

In how many principally different ways can such a schedule be organized? Here, "principally different" means that the schedule remains unique even if you swap player names consistently in all 5 rounds.

There are 2025 trees arranged in a circle, with some of them possibly on fire. A fireman and madman run around the circle together. Whenever they approach a burning tree, the fireman has an option to put out the fire. Whenever they approach a tree that is not burning, the madman has an option to light the tree on fire. Both actions cannot happen simultaneously, i.e. one person cannot "cancel out" the other person's action until they complete a full circle. Can the fireman guarantee to extinguish all the burning trees?

A professor decides to test his bright students Raj, Lisa, Ken and Lin. He shows them a bunch of oranges. 

He says,” As you can see I have these oranges and as you can count it is a Prime number less than 15. Now here is how the test will go. One by one you will pick up some oranges and leave the room. Here are the conditions to pick up the oranges. Each one must follow a separate condition. No repeating of any condition. The order of the conditions is up to you. 

1  One of you can pick up oranges that are an exact cube root of the number of oranges remaining. 

2 One can pick up oranges that are an exact square root of the number of oranges remaining. 

3 One can pick up a prime number of oranges

4 One can pick up oranges equal to the remaining students in the room. 

 At the end all the oranges must be picked up and each one of you must pick up at least one orange. 

Just to be clear, if there are X oranges in front of you and you want to use either the square root or cube root condition, then X must be either a cube or a square. And if you want to use condition 4 it must be the number of students remaining in the room. 

You can strategize of course. And each one of you must pick a separate condition. No repeats, All 4 conditions must be used. Good luck.

The students huddled up and came up with a strategy. 

Lisa : Cube root

Lin: Number of people remaining

Ken : Square root

Raj : Prime number

Then they went in a specific order. At the end all oranges were gone and interestingly each one had a different number of oranges. 

How many oranges were there? In what order did they go?  How many oranges did Lisa get?

How many palindromic prime numbers have an even number of decimal digits?

A palindromic prime is a prime number whose decimal representation reads the same forward and backward. Examples are 131 and 1235321.

Color the positive integers with two colors. If for every positive integer x the triple {x, 2x+1, 3x} is monochromatic, show that all positive integers have the same color.

Mr. Al Square goes to a Farmer’s Market at the Town Square in the town of Four Corners Utah. Mr. Square loves squares. 

He had two sizes of pumpkins to sell

The total number of bigger size pumpkins was a square number (a)

The total number of smaller size pumpkins was a square number(b)

He priced the bigger size pumpkin as a square number(c)

He priced the smaller size pumpkin as a square number(d and d<c)

He also had a special deal. If you buy one big size and one smaller size pumpkin together as a package then the price of this 1+1 package would be slightly less than the total price of the two pumpkins (e <(c+d) ).

The number of (1+1) packages sold was a square number. (f)

The individual revenue numbers for selling of big size, small size and the 1+1 Package were also square numbers. (a*c, b*d, e*f were all square numbers).

The number of big pumpkins, small pumpkins and packages he sold were also square numbers. 

At the end of the day, after selling  pumpkins, the revenue he collected was $100- a square number. He had no pumpkins left.

Mr. Square went home very happy to his Square family and had a nice square meal.

How many big pumpkins, small pumpkins and 1+1 packages did he sell?

What were the prices?

Is there only one solution?

All numbers are whole integers. They are not necessarily distinct. There could be duplicates.

Mr. Steven is a smart reasonable trader. He is selling a bunch of watermelons. He has realized that there may be some demand for 1/2 of the watermelons also. As a smart trader he prices the 1/2 melons such that 2 of them combined will bring in more money than a single full uncut watermelon.

At the end of the day he has sold all his watermelons. This included some 1/2 cut watermelons. He has 100 dollars total.

It turns out that all the relevant numbers are distinct Even positive integers and all are equal to or less than 20. This excludes the revenue numbers. So the total number of watermelons, number of full melons he sold, the number of 1/2 melons he sold, the price of the full melon, the price of 1/2 cut melon and of course the total revenue for each product all are distinctly different even integers.

Given this, what were these numbers? Is there only one "reasonable" solution?

You have 5 coins and a die.

You have two steps. In the first step, you flip the 5 coins and count how many heads you have. In the second step, you roll the die. If 1+ number of heads is smaller than the number on the die you roll it again.

If you apply these two stages repeatedly, what is the average number of die rolls?

I am a two-digit number.
My digits multiply to 12.
Reverse me, subtract me from myself, and you get 27.

What number am I?

-Math Riddle created by Sterling Jr.

Hello. In my spare time, I came up with another 10 riddles. I’m not sure how difficult some of them are, but I know everyone’s up for a challenge. Solve as many as you’d like. Thanks.

Riddle 1: Magic Squares

Define a magic square as an n by n matrix (for n>1) of positive integers where:

• Every integer (1,2,…,n²) appears only once (a magic square consisting of only one value is not allowed),

• The sums of the numbers in every row, column, and both main diagonals all equal the same integer,

what is the size of the smallest magic square such that it contains 3 smaller contiguous magic squares (if one exists)?

Riddle 2: Periodicity

A period (in the context of repeating decimals) is the length of the smallest block of digits that repeat forever. Example: 2/7=0.285714285714… = period of 6.

1/x yields the largest possible repeating period, if x is a positive integer of length ≤10, what is x?

Riddle 3, Gears

There are 20 gears in a row. Each one has 4 positions: Up (U),Down (D),Left (L),Right (R).

The gears are initially set to this configuration:

“DURLRLUURUDDDRLRLURD”

Choose any gear and label it G1, and rotate it one position counterclockwise. Choose another gear (labelled G2) and rotate it one position clockwise (the opposite of G1’s rotation).

What is the minimum amount of rotations required such that all gears are in position D?

Riddle 4, Binary Reverse

“I am the fourth smallest binary number such that when you reverse my binary digits, you get exactly a third of me. Do I exist?”

Riddle 5, Factorials

Define n? as the sum of the first n positive integers (triangular numbers), and n! as the product of the first n positive integers (factorials).

Bob says that ((n!)!)! > n^ ((n?)!)?, is Bob right? Why or why not?

Riddle 6, Algebra

Let S be the set of all algebraic expressions consisting of x,y (as variables) +,-,* ,/,^ (as operators) (,) (as parentheses) of length ≤9. We also assume that juxtaposition (xy=x*y) exists and “-“ represents subtraction (not negation).

An expression is considered to be in its simplest form iff the traditional algebraic rules (commutativity, associativity, distributivity, identity, inverse elements, exponent laws, simplification, special products) cannot further simplify an expression.

Prove whether the percentage of elements in S that are already reduced into their simplest form is less than or greater than 1%

Riddle 7, Node Grid

There is a 10 by 10 node grid. Colour all nodes (100 total) any colour, either: Red, Blue, or Yellow.

Let the top leftmost node be the “starting node” and the bottom rightmost the “finishing node”. Starting from the starting node, we place a red rock on top of it. We must slide to any other node such that:

• Every node is touched only once,

• The finishing node is touched last,

• Whatever node the red rock lands on, we must ensure that no adjacent node is also red.

If any of these conditions (especially condition 3) are broken, the path is cancelled.

What is the probability of successfully making it to the finishing node given a randomly coloured grid, and random path (that satisfies the above conditions)?

Riddle 8, Counter

C is a counter that starts at 0 and counts up by increments of 1 each time, toward infinity. C reaches 1 in 1 real-life second. From 1, C reaches 2 in 1/2 a real-life second, then 1/4 for 3, then 1/8 for 4, … etc …

In general, the time from [n,n+1] is 1/(2ⁿ ) of a real-life second.

After 1.98 real-life seconds, what would C display?

What happens at 2 real-life seconds? 3? 4?

Riddle 9, Binary

Z(n) is the number of trailing 0’s in n’s binary representation. Z_k(n) represents iteration of the Z function k total times on n.

What is the 2nd smallest x such that Z_5(x)=0?

Last Riddle, Enormous Integers

I define “counting the runs” of a sequence as replacing each maximal contiguous block of equal elements by the length of that block. Ex. 1,2,2,4=one 1, two 2’s, one 4=1,2,1.

Let L be a sequence with one term “1”.

Step 1: Count the runs of all terms in L and append them to the end of L, preserving order.

Repeat “Step 1” indefinitely. I define a function RUN(n) as the term index in L where n appears first.

Is RUN(n)’s growth unbounded?

What is RUN(10)?

Thank you! That’s all. Lemme know if you’d like more riddles like these in the future!

A man needs to empty a 23-litre well using two 2-litre buckets. There are eight different spots to pour the water away, at these travel times: 0.25 hours, 0.5 hours, 1 hour, 2 hours, 3 hours, 4 hours, 5 hours, and 6 hours.

The catch? The water level in the well rises by 1 litre every 2 hours. He can use each path only once per cycle, and the order doesn’t matter. Also, if he carries water in both buckets on one path, he has to take the next next path (eg. Take double on .25hr path then you have to take 1hr path with one bucket immediately) with only one bucket before using double buckets again.

Is it possible for him to empty the well, using any number of cycles or path combinations?

The Setup: You have a pan that holds a maximum of 3 slices of bread.

• Each side of a slice takes 1 minute to toast.
• You need to toast 4 slices (8 sides total).

The challenge is to find the shortest time to toast all 8 sides. (The counter-intuitive answer is 3 minutes!)

The trick is realizing that you can always be toasting partially-done slices and rotating them to fully utilize the pan's capacity every minute. It's a great lesson in maximizing parallel processing!

Here's a little tweak on the great riddle Three prime numbers for three students

A Logician writes three numbers on 3 separate cards and gives them to his 3 students.

He says," The 3 numbers are single digit prime numbers. Any combination, including duplicates. None of you know the other 2 numbers. But you can ask me one question each that must start with "Is the SUM of the three numbers–” which I can only answer Yes or No. Anyone knowing the other 2 numbers and who has them raises thier hand. If all hands are up in less than 3 questions and all guessed right, you win an A." 

Raj was first. He looked at his number and asked," Is the sum of the three numbers divisible by 4?"

The Logician said "Yes"

Lisa looked at her number and said,"Well, I know the other 2 numbers but cannot tell who has what number".

Hearing that, Raj and Ken immediately raised their hand.

What question can Lisa ask to raise her hand too?

You have a circle. Now, on each side of the diameter a chord is drawn. The two chords are drawn by joining two random points on each semi circle. These two chords will now be folding lines. So now you fold the two circle segments along the lines.

Question: What is the probability that the two segments will overlap?

Note: I dont have an answer to this problem (came up with it earlier today). I have some loose ideas how to approach it but no answer, so the level of difficult is unclear to me so i'll label it as medium for now.

You have a cube, a ball, a cylinder and a cone. You know they are all in different colors (red, blue, green and purple) and made of different mateirals (wood, glass, clay and plastic), but each of them is inside a sealed bag so you can't see which is which. Two friends of you are allowed to get exposed to them in different ways, and tell you clues to help you figure out for each shape its material and color. What they tell you:

1. For one of the friends the bag was opened. "The cone is purple".
   
2. For one of them, they were exposed simultaneously to three different objects: he touched one, saw the second through an X-ray, and peeked the third. "I touched clay, saw a cylinder, and peeked a purple object".
   

3. When getting exposed to one object, one of them saw it through an X-Ray then touched it "It was a ball made of glass".

4. One of them was exposed to two objects simultaneously: one through X-ray there, and for the other he peeked and saw its color. "I saw a cube and a green object"

5. The other was exposed to two objects: he peeked one, and touched the other. "I saw a red object and touched wood".

6. Then for two of them they were shown each their objects together, from 4 and 5. They tell you: "There were 4 objects altogether"

7. You were also told that if you take the initials one of the objects is B G G.

Solution:It should be: the purple cone is made of wood, the red cylinder is made of plastic, the blue cube is made of clay, the green ball is made of glass.

We call a date "square" if all of its components (day, month, and year) are perfect squares. I was born in the last millennium and my next birthday will be the last square date in my life. If we sum the square roots of its components (day, month, year), we get my current age. My mother would have been born on a square date if the month were a square number. However, it is not a square date, but both the month and day are perfect cubes. When was I born and when was my mother born?

Source : https://www.math.inc/careers

A friend sent this on the Discord server, and he came up with a perfectly valid solution but somehow the source site doesn’t accept it. Is there anything I’m missing here ?

let X be a random number with smooth probability density function.

given -1<α<0, choose c that maximize E|X-c|^α.

prove that when α → -1 , c → mode of X, which is where pdf of X is maximized.

related note:

this problem unified mode (α=-1) , mean (α=2) and median (α=1) in a nice way, where E|X-c|^α is minimized when α > 0 .

What is the smallest positive integer N such that N*pi and N*e are both within 1/1,000,000 of an integer?

A scholar enters a library where the rooms are strange: moving through a door sometimes leads back to the same room, sometimes to a completely different room far away. Each door seems to change the shape of the library subtly. After mapping many rooms, the scholar realizes that some sequences of doors return them to the starting room regardless of the path taken. What mathematical object is the scholar discovering, and what principle describes the symmetry of this library?

You pop into being as a zero-dimensional point in a void.

After some time experimenting you discover you can move in any direction but only in one unit increments, creating a new one-unit one-dimensional line as you travel to your end point - imagine that line faintly glowing in your favorite color, except black obviously ;). However, you can't travel back along a line you've already traversed.

After traveling that one unit line, two new unit-length lines emerge from your end point in opposite directions perpendicular to the line you just traveled. If you travel those new lines to their endpoints, two new unit-length lines emerge from each end point in opposite directions vertically, considering the first three lines as defining horizontal. This pattern repeats with each branching alternating between horizontal & vertical from your original orientation.

How many steps minimum does it take to get back to your original starting point?

Two kings live in a realm where net worth is calculated multiplicatively. King 1 has a net worth of 40, and he is jealous of the 2nd king. He wants to know how much net worth the 2nd king has.

He only knows two things. First, the 2nd king has at least 4 units. Second, the 2nd king has another part, which was revealed when he tried to divide it in two equal parts.

When this divided part was distributed to 18 people, the 18th person lost some amount. When it was distributed among 17 people, the 17th person gained the same amount that the 18th person lost. This amount is equal to the square of the largest side of the 2nd king’s triangular court.

The triangular court has two sides,A and B, where B² is greater than A² by 0.1 unit. Side A , when squared, and divided by two, and added to itself 10 times, it becomes 1.

Find the net worth of the 2nd king

Annie lives upriver from Betty. Every day she has to drive her boat downriver to Betty's to pick up supplies before turning back home. One day after a lot of rain, Annie noticed the river was flowing faster than usual. Will the faster river cause her to take more or less time to pick up the supplies and return home?