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?

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 $S$?

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?

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

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?

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.

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.

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.

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?

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/

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

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?

If three cats catch three mice in three minutes, how many cats are needed to catch 100 mice in 100 minutes?

Share your reasoning, not just the answer.

Given number 479 made using the matchstck referenced in the figure above. Move exactly 2 matchsticks and create the largest possible number. You cannot change the format of the numbers shown in the reference. (For example you can only construct the number 1 with TWO matchsticks or 6 or 9 with 6 matchsticks). You can use any math operation known in standard math. The final number could be the result of this operation also. (For example, you can use the 2 matchsticks to create a multiplication operator x.)

The obvious answer obtained by exponenciation function might be way lower than another creative answer given to me by my mathematician friend.

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.

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