Puzzles, curiosities, brain teasers and other mathematics done "just for fun".

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1
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1answer
118 views

what is the formula for determining the next year in which a given month/day will occur on a specific weekday

So, I was trying to express the formula for determining the next year on which a given date (month/day) will fall on a given weekday. The internet has plenty of sites that explain how to determine ...
1
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2answers
330 views

maximum number of independent bishops on a nxn chessboard

So I came across this problem where we have to find the maximum number of independent bishops on a nxn chessboard such that no two bishops attack each other . So after drawing the cases for $3$x$3$ , ...
3
votes
1answer
110 views

Proof involving an isosceles triangle

I came across this problem in some (maybe) high school book: Let $ABC$ be an isosceles triangle s.t. $AB=AC$. Also, $\alpha>\beta$. It is known/given: ...
2
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1answer
2k views

How many bit strings of length 15 have exactly three 0s?

I need help with this question: How many bit strings of length 15 have exactly three 0s?
0
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1answer
137 views

An equation of the form A + B + C = ABC

So I was on a SPOJ spree until I came across this question . The question says $$\tan(\frac{1}{A}) = \tan(\frac{1}{B}) + \tan(\frac{1}{C})$$ where we have to find the $\min(B+C)$ for a fix $A$ where ...
2
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2answers
105 views

What is the Smallest Integer $N$ Where Reversing the Digits Makes $3N$?

What is the smallest positive integer N such that the integer formed by reversing the digits of N is triple N? (Does such an integer even exist? If not, then for what multiplier for $N$ will such an ...
18
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2answers
3k views

Interview puzzle with a deck of cards, some cards upside-down

You are sitting in a dark room. It is completely dark. You can't see anything and there is no way that you can make light. Basically, just assume that you are blind for this task. There is a table in ...
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0answers
315 views

Proof of convergence of Kaprekar's Constant

I've tried googling this one a bit but nothing seems to come up, even though its considered to be a well known fact. Why does the kaprekar process of taking a 4 digit number: L, generating L' and L'' ...
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2answers
95 views

how to find taxicab numbers but for squares?

Natural numbers that can be written as the sum of squares in two or more ways. The first ten numbers are 50, 65, 85, 125, 130, 145, 170, 185, 200, 205. $$ n = a^2 + b^2 = c^2 + d^2\\ a^2 − c^2 = d^2 ...
4
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1answer
135 views

Factorial of Complex Values

Since the gamma function is an analytic continuation of the factorial function, we can find the factorial of complex values. How does one go about doing so? I've looked far and wide on the internet ...
22
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1answer
1k views

Mathematical Intuition Behind Schizophrenic Numbers?

Schizophrenic numbers (A014824) are numbers whose square roots "look" like rational numbers. They were first discussed in 2004 by Darling in the Universal Book of Mathematics (page 282), and I ...
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2answers
717 views

Breaking chocolate bars game

About two weeks ago, a friend of mine taught me the following game without his knowing the answer. It may be famous, but I haven't known it. There are $N\ (\in\mathbb N)$ chocolate bars composed of ...
1
vote
1answer
127 views

total number of combinations?

Patient Age ---> Avg Visits / Year <1 year ---> 7.5 1-4 years ---> 3.0 5-14 years ---> 1.8 15-24 years ---> 1.7 25-44 years ...
12
votes
1answer
600 views

Minimum and maximum determinant of a sudoku-matrix

Let $A$ be a sudoku-matrix. Assume that its determinant is positive. What is the lowest, what the highest possible value for the determinant of $A$ ? $A$ must have the dominant eigenvalue $45$, but ...
6
votes
3answers
400 views

$\pi$, $e$, $\phi$, and sunflowers

While reading some internet materials on design, I came across this picture and comment: I found it a little bit surprising. I knew that the real sunflower follows golden ratio in some way (but I ...
7
votes
2answers
367 views

Evenly space holes in circle

A picture is worth a thousand words: This gear is part of an interactive SVG Spirograph I'm creating. I'm dynamically generating the gear based on a number of parameters (gear radius, number of ...
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2answers
198 views

What is the probability of going bankrupt in roulette?

Imagine that the bank has the money $M_1$ and the player has the money $M_2$. The rules are the following: You win with a chance of $\frac{17}{36}$ and lose with $\frac{19}{36}$ each round. Now you ...
4
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0answers
142 views

$Z_n \backslash \{0\}$ splits into octets

Let $n=8m+1, m\in\mathbb{N}$. Does the set of nonzero elements of $\mathbb{Z}_n$ split into disjoint octets of the form $8_k=\{\pm a_k,\pm b_k,\pm a_k\pm b_k\}$? The computer tells me it is possible ...
3
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2answers
357 views

Winning a card game

Scenario: Each player has a deck of N cards. The first player controls an object called Grindclock, which means that at each turn, he can either : Add a "charge" counter on Grindclock, or Remove the ...
2
votes
3answers
121 views

What are the properties of the set of the Real Numbers without the Integers?

This question came up in a lunchtime discussion with coworkers. None of us are professional mathematicians or teachers of math. I apologize for any incorrect math or sloppy terminology. We were ...
2
votes
2answers
372 views

Pirates And Coins No.1

I actually like this one: There are five pirates in a ship and they have found 100 coins. The biggest pirate offers a way to divide the coins. If at least half of them agree on the division, it will ...
1
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2answers
1k views

Number of Ways to Draw a Pair in a Poker Deck

I am confused over calculating the number of ways in which I can select a pair out of a deck of 52 cards. This is how I go about solving the problem: Following the definition of a pair in card ...
0
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2answers
120 views

Fun Tan Question [duplicate]

Using only trig identities, how would you approach the following question? Determine the value of $$ \prod_{i=1}^{89} \tan i^° = \tan 1^° \cdot \tan 2^° \cdots \tan 89^° $$
0
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2answers
130 views

time and distance

Dexter and Prexter are competing with each other in a friendly community competition in a pool of 50m length and the race is for 1000m. Dexter crosses 50m in 2 min and Prexter in 3 min 15 sec. Each ...
2
votes
4answers
316 views

What is the sum of the numbers in the shaded circles?

Each of the 6 circles contains a different counting number. The sum of all 6 numbers is 21.The sum of the 3 numbers along each side of the triangle is shown in the diagram. so What is the sum of the ...
0
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0answers
59 views

Card Shuffling and Convergence in Probability

There are $4n$ cards, and we denote the set of cards with number $4k,k \in \{1,2,\ldots,n\}$ as $S$. The we shuffle the whole cards randomly, which means that each permutation will happen with the ...
0
votes
1answer
202 views

How many times can you round a number?

For a typical rounding algorithm, I'm wondering how long the rounding chain goes for when you round up a number. For example, if you have a decimal like 0.4445, you round the last 5 up, which would ...
1
vote
1answer
22 views

Creating switches for piece wise defined function?

How can I create "switches" [the term may be new, but I'll explain it] for piecewise defined functions ? Suppose a function: $$ f(x)=\begin{cases}\alpha\;,x\in D_1\\\beta\;,x\in ...
-1
votes
1answer
236 views

What proportion of the circle is covered by the square?

Or what is the combined area of the circle segments (chords)? Picture a circle which is covered by a square, where the bottom vertices of the square are inscribed within the circle (so that the ...
6
votes
1answer
83 views

Sparsest matrix with full inverse

What is the sparsest matrix in $\mathbb R^{n,n}$ such that the inverse is full? I.e. I am looking for a matrix $A\in \mathbb R^{n,n}$ with as few non-zero entries as possible, such that $A^{-1}$ has ...
0
votes
2answers
129 views

Wine problem - ratio and mixture

Question $8$ litres are drawn from a cask full of wine and is then filled with water. This operation is performed three more times. The ratio of the quantity of wine now left in cask to that of the ...
11
votes
2answers
605 views

Is 39 moves the longest a chess game can go moving only pawns?

I've thought of a few different ways a chess game could go on moving only pawns, but I've only counted moves in one scenario: Both White and Black take 16 moves to line their pawns at the middle of ...
0
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2answers
79 views

Questions about area

In math class (I'm in Geometry) I was messing around and decided to try and find the area of a circle using the area of a square if the radii are the same length. The square is inscribed in the ...
3
votes
1answer
72 views

How many orientations are there for pawns (for a single player) on a chess board?

How many orientations are there for pawns on a chess board? Pawns can only move forward or diagonally forward. No two pawns may exist on the same square. Pawns start on the second row, and ...
0
votes
2answers
192 views

Is my solution of this time and distance problem correct or wrong?

P and Q start running in opposite directions (towards each other) on a circular track starting at diametrically opposite points. They first meet after P has run for 75m and then they next meet after Q ...
0
votes
3answers
174 views

what will be the answer either 30 or 31 for this time based question?

A red light flashes 3 times per minute and a green light flashes 5 times in two minutes at regular intervals. If both lights start flashing at the same time, how many times do they flash together in ...
2
votes
1answer
92 views

How to use the Lambert W function in an equation like this?

I was thinking of $\frac{4}{3}$ and found that $(\frac{4}{3})^4$ roughly equals $3$ (very roughly), and I though I would try to find the pairs of numbers where the equality suffices. So given the ...
1
vote
1answer
76 views

minimize distance

consider a two dimensional system. two points are given whose co-ordinates are $(h1,h2)$ and $(k1,k2)$. I want to minimize the distance between these two points with the condition that person has to ...
12
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0answers
1k views

This should be a piece of cake… right?

You probably know the following problem: We have two circular cakes of the same height but unknown and potentially different radii, and we want to cut them into two equal shares. Each cut can only ...
4
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0answers
142 views

A game on a smaller graph

In this question Alice and Bob play a game on $K_{2014}$, Alice directing one edge, Bob directing $1$ to $1000$ edges with Alice trying to make a cycle. The proof that Alice can win depended on the ...
27
votes
2answers
1k views

The expected outcome of a random game of chess?

Imagine a game of chess where both players generate a list of legal moves and pick one uniformly at random. Q: What is the expected outcome for white? 1 point for black checkmated, 0.5 for a ...
1
vote
1answer
140 views

Magic square of numbers

What is the logic behind filling up a magic square? I have understood the algorithm of filling up a magic square of 3*3 or 5*5. I really do not know how to derive this
1
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1answer
3k views

logic\math question [closed]

Hi, This is allegedly a very simple question....most of the people answered that the answer is 9=90. i claim that theoretically, you can't be 100% sure about the right answer because you don't know ...
1
vote
1answer
186 views

How is this card/paper plate magic trick done?

There are 3 paper plates, in which A is written on one, B on the 2nd, and C on the third. Now the person performing the trick knows the initial order of the paper plates. He then asks an audience ...
0
votes
2answers
455 views

A simple Question. How much “faster” is A than B as a percentage figure?

A simple question If two people complete the same task, and Person A completes the task in 10 minutes, Person B in 8 minutes, what figures can/should be quoted in terms of how much quicker A is than ...
2
votes
1answer
90 views

Polynomial equal to the ceiling of x

For a few days, I've been looking for a polynomial who's value is equal to the ceiling function of the only variable it contains. I thought about it for while and I haven't got a clue.
5
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0answers
147 views

Interesting Number Game

Let's say you are given some random positive integer. Next, assume you are given the following sequence of integers: $$[1, 2, 3, 4, 5, 6, 7, 8, 9].$$ Now, assume you are allowed to use the following ...
0
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2answers
39 views

IS the sequence right?

What i think is that, the sequence isn't correctly formulated. I have tried like this 7x5-9=35 2x3-1=5 8x6-4=44 4x5-9=?=11
1
vote
2answers
921 views

Concept of Probability in math first level

I am trying to teach myself the concepts of probability and I was wondering if this is correct. I am only 13 years old and did not learn this yet. I am just reading parts of a probability book to get ...
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1answer
1k views

Mean and Standard Deviation self thought problem

I am 13 years old trying to teach myself about standard deviation and was wondering how this problem would look like. I know I am young to be learning this but I was reading about this and got ...