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

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49 views

Primes made from sequential digits

While messing around, I noticed that across some prime numbers contain only sequentially increasing digits, e.g. $23, 67, 89,23456789$. If we adopt a convention of returning to $1$ after a $9$, we ...
9
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2answers
170 views

Good Reference for Justifying (less well-known fields of) Math?

How do we as mathematicians justify the study of math to students? Or, indeed, how do we justify it to the general public? How do you justify your particular field? I'm particularly interested in ...
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2answers
58 views

4 crystal balls and a 10,000 story building

There is an analog of this question I've heard with 2 crystal balls but a higher number like 4 or more makes it much more interesting. You are given 4 crystal balls and there is a 10,000 story ...
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3answers
127 views

Blue Eyes: A Logic Puzzle, has a puzzling solution (a.k.a. What does common knowledge have to do with it?)

In Blue eyes: a logic puzzle (specifically, the follow up questions), the most common answer is that it needs to be common knowledge that someone has blue eyes for all the blue-eyed people to leave. ...
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7answers
2k views

Mathematical literature to lose yourself in

H.M. Edwards in the preface to his book on the Riemann Zeta Function, summarises his philosophy on learning Mathematics: ...I have tried to say to students of mathematics that they should read the ...
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2answers
82 views

Natural numbers verifying $P(n) = n^2 - 42n + 440$, where $P(n)$ is the product of the digits

Let $P(n)$ be the product of the digits of the number $n$, with $n \in \mathbb{N}$. What is the product of all the natural numbers $n$ that verify the equation $P(n) = n^2 - 42n + 440$? I ...
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1answer
248 views

Joke explanation: “a comathematician is a device for turning cotheorems into ffee”

Ok, so apparently there is an old joke (which I DO get) that says that in Hungary a mathematician is a device for turning coffee into theorems. I found a post by Qiaochu Yuan that has the following ...
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1answer
33 views

Csn someone produce a sudoku puzzle where guessing more than one cell's value at a time is required?

Currently I have a sudoku puzzle solver program and I've tried all the puzzles I can find that are labeled the "hardest" on various sudoku video games and puzzle books. My solver has solved them all. ...
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80 views

Binomial triplets

Solutions to the equation $$\dbinom{a}{n}+\dbinom{b}{n}=\dbinom{c}{n}$$ I will refer to as 'Binomial triplets of order $n$'. These triplets describe simplicial $n$-polytopic numbers that can be ...
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1answer
100 views

Can 23 of this polycube fit in a 5x5x5 box?

Consider the following pentacube (front and back view shown.) I have used Burr Tools to determine that 24 of these will NOT fit in a 5x5x5 box. According to my notes when working on this problem a ...
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1answer
235 views

Which is greater, $20 \uparrow\uparrow\uparrow\uparrow 20$ or $4 \uparrow\uparrow\uparrow\uparrow\uparrow 4$?

This past Wednesday's What-If had this image at the bottom: In particular, I am interested in $20 \uparrow\uparrow\uparrow\uparrow 20$. I immediately thought of Graham's Number, but clearly that ...
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1answer
47 views

Why does the filling up of odd order magic square with numbers follow the knight movement?

Why does the filling up of odd order magic square with numbers follow the knight movement? I was reading about magic square, where I came up with the knight movement filling up of the magic square ...
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1answer
70 views

How can “Lucky Numbers” be approached rigorously?

To begin with, "Lucky Numbers" are a sequence of numbers generated by a sieve similar to the Sieve of Eratosthenes for finding primes. It starts with the set of natural numbers. Begin by selecting ...
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2answers
86 views

Prove withoui calculus: the integral of 1/x is logarithmic

It was known in the 17th century that the function $$ t \mapsto \int_{1}^{t} \frac{dx}{x} $$ is logarithmic: a geometric sequence in the domain produces an arithmetic sequence in the codomain. This ...
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1answer
87 views

2048 algorithm for merging

Ok, here's a question my friend just sent me, ive mastered it to some extent, but am failing, so, please help a little: Your target is to merge these blocks in such a way that one bigger number is ...
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2answers
31 views

Suppose T(k) denotes the smallest number of steps needed to move from k to 100.Find y & z such that T(9)= 1+ min (T(y),T(z)).

Suppose you want to move from 0 to 100 on the number line. In each step, you either move right by a unit distance or you take a shortcut. A shortcut is simply a pre-specified pair of integers i,j ...
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1answer
42 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 ...
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2answers
61 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$3$ , ...
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0answers
42 views

Is it possible to calculate how many people pay full price from the following numbers?

I'm currently analysing the Activision Blizzard earnings call for Q2 2014 and 2014 to see if I can figure out how many North American and EU subscriptions there are of the 6.8 million World of ...
3
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1answer
32 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: ...
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1answer
111 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?
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1answer
97 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 ...
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2answers
92 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 ...
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2answers
2k 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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1answer
69 views

How much distance did messenger cover? [closed]

A column of troops $80$m long is moving along a straight road at a uniform pace. A messenger is sent from the head of the column, delivers a message at the rear of the column and returns. He also ...
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0answers
62 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
78 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 ...
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0answers
59 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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1answer
433 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 ...
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1answer
83 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 ...
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2answers
130 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 ...
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2answers
104 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
110 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 ...
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0answers
123 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
165 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 ...
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3answers
99 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 ...
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2answers
339 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 ...
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0answers
91 views

An algorithm for linear equation system problem

Is there an algorithm for the following: I have 30 linear Diophantine equations of the following form: $$a_{1,1}x_1+\cdots +a_{1,16}x_{16}=b_1$$ $$a_{2,1}x_1+\cdots +a_{2,16}x_{16}=b_2$$ ...
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0answers
40 views

How to calculate the number of combinations of getting a pair in a deck of 52 cards?

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 games, ...
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2answers
102 views

Fun Tan Question [duplicate]

Using only trig identities, how would you approach the following question? Determine the value of $$ \prod_{i=1}^{89} \tan i^° = \tan 1^° \cdot \tan 2^° \cdots \tan 89^° $$
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2answers
68 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 ...
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4answers
90 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 ...
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0answers
40 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 ...
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1answer
66 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 ...
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1answer
19 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 ...
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1answer
98 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 ...
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1answer
60 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 ...
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2answers
53 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 ...
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0answers
13 views

Is this a good formalization of self-counting functions?

I have looked on the internet, and I do not see any formal definition of a self-counting function. So I offer my own. The question I put to you is whether this is a good formalization . We have a ...