0
votes
0answers
37 views

Using jugs filled with water problem

Given jugs $m$ and $n$ liters (WLOG $m<n$) is it always possible to get all $i$, $0 \leq i \leq n ?$ If so, prove it. If not, explain which $i$ you can get. Is there also a minimum number ...
4
votes
0answers
94 views

Adding Numbers Pattern

A few nights ago I couldn't sleep and so started doing this: I would take a number and add up all of its digits to get a new number and then add up all of those digits and so on until there was only ...
59
votes
12answers
14k views

Dividing 100% by 3 without any left

In mathematics, as far as I know, you can't divide 100% by 3 without having 0,1...% left. Imagine an apple which was cloned two times, so the other 2 are completely equal in 'quantity'. The totality ...
0
votes
1answer
22 views

Condition for L and R to make the below scenario true?

You are a given a number N. N <= 10 ^ 9 . You are given a range of numbers L and R . ...
2
votes
2answers
60 views

Show that 4*AB=CAB is not possible

Show that 4*AB=CAB is not possible.Each letter denotes a single digit.It could be noted that since LHS is a multiple of 4,thus RHS would also be a multiple of 4.That's all i conclude yet...
4
votes
1answer
165 views

A 5x5 board has 25 cells.The numbers $\{1,2,3,4,5\}$ are written on every row,every column and the two main diagonals.

A 5x5 board has $25$ cells. The numbers $\{1,2,3,4,5\}$ are written on every row,every column and the two main diagonals without any repetition. If the sum of the numbers of the diagonal below the ...
3
votes
2answers
124 views

Number-Theoretic Coin Puzzle

There are three piles of coins. You are allowed to move coins from one pile to another, but only if the number of coins in the destination pile is doubled. For example, if the first pile has 6 coins ...
3
votes
1answer
249 views

Find a number leaving a particular remainder with 3 different numbers

I have the following question: Let $N$ be the greatest number that will divide $1305, 4665$ and $6905$, leaving the same remainder in each case. What is the sum of digits of $N$. My approach ...
0
votes
3answers
78 views

Find the digits $A,B,C$ such that $ABC+BAC+CAB=ABBC$

A,B,C are distinct digits of a three digit number such that ...
3
votes
2answers
129 views

The number 3211000 is 7-special

Define a positive integer $k$ to be $n$-special if it satisfies the following properties: It has $n$ digits (0, 1, ..., 9) The 1st digit is equal to the number of 0's in the decimal representation ...
6
votes
2answers
234 views

$20$ hats problem [duplicate]

I've seen this tricky problem, where $20$ prisoners are told that the next day they will be lined up, and a red or black hat will be place on each persons head. The prisoners will have to guess the ...
0
votes
1answer
125 views

Pascal's other triangle

Just a brainteaser question: Can you identify the generator of the following pattern of numbers?      Remark on any interesting patterns you see in the triangle.
1
vote
2answers
1k views

Counting squares of maximum size in a rectangle

Given a rectangle of sides $m$ and $n$. $( m,n \in [1,1000] )$ We can cut the rectangle into smaller identical pieces such that each piece is a square having maximum possible side length with ...
5
votes
2answers
143 views

When is $\frac{2^n+1}{n^2}$ an integer? [duplicate]

Can anyone see how to solve this number puzzle? Find all integers $n>1$ such that $$\frac{2^n+1}{n^2}$$ is an integer.
4
votes
2answers
59 views

Find $x,y$ such that $x=4y$ and $1$-$9$ occur in $x$ or $y$ exactly once.

$x$ is a $5$-digits number, while $y$ is $4$-digits number. $x=4y$, and they used up all numbers from 1 to 9. Find $x,y$. Can someone give me some ideas please? Thank you.
4
votes
4answers
708 views

Number system - sum of two digit numbers

The sum of four two digit numbers is $221$. None of the eight digits is 0 and none of them are same. Which of the following is not included among the eight digit ? $$(a) \;\;1 \\ (b)\;\; 2 \\ ...
4
votes
4answers
469 views

How to calculate the number of pieces in the border of a puzzle?

Is there any way to calculate how many border-pieces a puzzle has, without knowing it's width-height ratio? I guess it's not even possible, but I am trying to be sure about it. Thanks for your help! ...
1
vote
1answer
450 views

Gödel, Escher, Bach: $ b $ is a power of $ 10 $.

I’d like to verify if my formula correctly expresses that a number is a power of $ 10 $, using the $ \sf{TNT} $ language provided by Hofstadter in his famous book Gödel, Escher, Bach: An Eternal ...
5
votes
2answers
131 views

How many cups of sugar do I need for these 5th grade problems?

Problem 1a: If 4 glasses of a mixture needs 1 cup of sugar how many cups of sugar are needed for 5 glasses? This one is easy and makes sense. It's just simply $\frac{1}{4}*5$ Now taking it a notch ...
0
votes
3answers
61 views

Problem related to a given diagram

I came across the above problem but do not know how to tackle it. Can someone point me in the right direction? Thanks in advance for your time.
1
vote
1answer
117 views

Explanation for a peculiar property of a number

I had come across a problem, where 2 people play a game where think of a number n, and turn by turn subtract a number $p$ from $n$ where $p$ is a prime and is $p < n$ and 1 is taken as prime here. ...
11
votes
1answer
677 views

Given a set of digits, what is the biggest number we can make using exponentiation - numberphile noodle quiz

The question is motivated by a question on a can of number noodles. Each item is a digit between $0$ and $9$. Clearly, if you form a string and consider it to represent a base $10$ integer, then ...
1
vote
1answer
486 views

How to solve multiplication alphametics?

I am referring to puzzles like these, where every letter represents a unique number (0-9): ...
1
vote
2answers
2k views

How to solve this alphametic (verbal arithmetic)?

I know I can get the answer for this puzzle but I'm struggling to see how to solve it. Every letter represents a different number (0-9): ...
2
votes
1answer
399 views

Guess my birthday

This is just a funny question that I was elaborating... I know one way to solve (or maybe it's wrong...), but I want know if there is another way to solve this (when we keep adding conditions, there ...
4
votes
3answers
833 views

Some digit summation problems

What is the sum of the digits of all numbers from 1 to 1000000? In general, what is the sum of all digits between 1 and N? f(n) is a function counting all the ones that show up in 1, 2, 3, ...
7
votes
1answer
192 views

Flirtatious Primes

Here's a possibly interesting prime puzzle. Call a prime $p$ flirtatious if the sum of its digits is also prime. Are there finitely many flirtatious primes, or infinitely many?
3
votes
1answer
232 views

Divisor/multiple game

Two players $A$ and $B$ play the following game: Start with the set $S$ of the first 25 natural numbers: $S=\{1,2,\ldots,25\}$. Player $A$ first picks an even number $x_0$ and removes it from $S$: ...
2
votes
3answers
273 views

Smallest $k$ s.t. $7x+1=9y+2=11z+3=k$, all positive integers

Find the smallest positive integer, which on dividing with 7 gives remainder 1, on dividing with 9 gives (remainder) 2 and that after division by 11 yields 3 as remainder. i.e., find smallest $k \in ...
2
votes
0answers
83 views

When does $n^2$ divide $2^n+1$? [duplicate]

Possible Duplicate: How many rationals of the form $\large \frac{2^n+1}{n^2}$ are integers? A friend of mine asked me this question over lunch, and it's been a week that I can't do ...
3
votes
2answers
308 views

puzzle about array of numbers

Consider an array of numbers $$ \color{#C00000}{1}\ \hphantom{7\ 6\ 5\ 4\ 7\ 3\ 5\ 7\ 2\ 7\ 5\ 3\ 7\ 4\ 5\ 6\ 7\ }\color{#C00000}{1}\\ 1\ \hphantom{7\ 6\ 5\ 4\ 7\ 3\ 5\ 7\ }\color{#C00000}{2}\ ...
159
votes
3answers
7k views

How many fours are needed to represent numbers up to $N$?

The goal of the four fours puzzle is to represent each natural number using four copies of the digit $4$ and common mathematical symbols. For example, $165=(\sqrt{4} + \sqrt{\sqrt{{\sqrt{4^{4!}}}}}) ...
1
vote
1answer
157 views

A rule to determine the crossed out digit

Lets take any integer, $z=abc\cdots$, form the sum of its digits, $a+b+c+\cdots$, subtract this from $z$, cross out any one digit from the result, and denote the sum of the remaining digits by $w$. ...
6
votes
1answer
151 views

Puzzle: Can you find an elementary proof that every $n \gt 6$ can be represented as a sum of $O(\log n)$ distinct primes?

Can you find an elementary proof that every $n \gt 6$ can be represented as a sum of $O(\log n)$ distinct primes? For example, $11 = 11$, $12 = 5 + 7$, $13 = 2 + 11$, $14 = 2 + 5 + 7$. On the other ...
4
votes
1answer
223 views

Expressing any given number in the form of $x^y + y^x$

I was told by one of my friends that any given positive integer can be expressed in the form of $x^y + y^x$ where x & y are integers. For example: 17 = $2^3+3^2$ Surprisingly,this could be done ...
27
votes
1answer
661 views

A fun Pascal-like triangle

Inspired by Pascal, I put on some shackles and a thorny belt. Inspiration came pouring in, and I thought of the following triangle: $$ \begin{array}{rcccccccccc} & & & & ...
15
votes
2answers
768 views

Puzzle: $(\Box @)+(\Box @) = (\Box\bigstar\Box$)

Some ETs follow a positional number system, with the same base as the number of fingers on their hand. The following inscription is all the evidence we have: $$(\Box @)+(\Box @) = \Box\bigstar\Box ...
4
votes
1answer
154 views

Specific ten digit number

I'm trying to solve this little problem. So far no luck. Could anyone help? Thanks in advance :) What is the ten digit number such that the i-th digit is the number of i's in the number ( 0<= i ...
4
votes
6answers
737 views

A tedious puzzle (but not homework)

There are 1000 light bulbs and 1000 tutors. All light bulbs are off. Tutor 1 goes flipping light bulb 1,2,3,4... tutor 2 then flips 2,4,6,8...tutor 3 then 3,6,9...etc until all 1000 tutors have done ...