# Questions tagged [number-theory]

A mathematical puzzle whose solution is heavily based on the arithmetic properties of the integers. General number theory questions are off-topic but can be asked on Mathematics Stack Exchange.

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### Find the value of $\bigstar$: Puzzle 10 - Uncertainty

This puzzle replaces all numbers with other symbols. Your job, as the title suggests, is to find what value fits in the place of $\bigstar$. To get the basic idea, I recommend you solve Puzzle 1 ...
617 views

You are given several pieces of paper which are as follows: (Unfortunately, a textual rendering is very difficult with this puzzle, so if someone can offer a suggestion on how to do it, that would be ...
156 views

### 3 bags of coins, find which bag contains fake coins, doing only 1 weighing

Probably a very easy puzzle for you guys; my point is more a mathematical one. FYI, it was shown in an episode of Columbo. You have 3 bags of coins. Each bag contains 50 identical coins. Two bags ...
79 views

### Hitting a car with a bullet [duplicate]

1) There is a road on which a car starts with an integral speed towards the left or the right, starting from an integral point (take the road to be the x-axis) 2) The speed and the point from which ...
427 views

### Smallest prime number which when spelt out contains the letters P, R, I, M, E

So inspired by recent slew of questions based on prime numbers. What is the smallest prime number when written out (Using the Western numbering system and English) would you encounter the letters P, ...
3k views

### Smallest PRIME containing the first 11 primes as sub-strings

In Smallest number containing the first 11 primes as sub-strings, @Alconja successfully found the smallest number which contains the first eleven primes (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31) as ...
705 views

### Sharing cake among 9 or fewer people

You are expecting guests to your birthday party. You know that there will be at most 8 guests, but you don't know how many will actually come. What is the smallest number of pieces you should divide ...
746 views

### The Royal Mint of Alphagonia

In the Kingdom of Alphagonia nothing can be bought for less than 30 alphas, the local currency. 1) What three denominations of coins should the kingdom mint so that as many as possible of the (...
549 views

### Winning the Lottery

Bob: I hear you won the lottery. Alice: So I did! Bob: What six numbers did you win it with? Alice: Can't remember. All I recall is that they were all different, and none greater than 28. Bob: ...
547 views

### National Graph Lottery

In UK's National Lottery players choose 6 different whole numbers in the range 1 to 59, and win a large prize if all six match with the day's draw. Each choice of six numbers by a player gives rise ...
587 views

### A Magic Flying Saucer

Place 19 different positive integers on the vertices of this graph so that the 13 products of three numbers in a straight line are all equal. Do so in such a way that the product is as small as ...
2k views

### A Magic Diamond

Place 15 different positive integers on the vertices of this graph so that the ten products of three numbers in a straight line are all equal.
231 views

### What is a Freecell Word™?

This is in the spirit of the What is a Word/Phrase™ series started by JLee with a special brand of Phrase™ and Word™ puzzles. If a word conforms to a special rule, I call it a Freecell Word™. Use the ...
902 views

### Box of tablets, whole or broken: solution required

This is a puzzle that I thought up whilst taking a course of meds. I currently haven’t solved it, and would be curious to know if anyone has a solution for it. Here goes: Scenario: John has a box of ...
147 views

### Positioning cards labeled with numbers from 0-9

Ann and Bob play a game. On a table there are 10 cards which are labeled with number from 0 to 9 each. Bob is allowed to change the position of the cards with a sequence of his preference. When he is ...
123 views

### A pile of chips involving powers of 2

Ann and Bob play alternately on a pile of chips. On each play, any number of chips, which is a power of 2 (including 1=$2^0$), can be removed from the pile. Obviously the number of chips to be removed ...
175 views

### A pile of chips involving primes

Ann and Bob play alternately on a pile of chips. On each play, either 1, 2 or 3 chips can be removed except if the number of chips is a prime number. In that case either 1, 2, 3, 4 or 5 chips can be ...
183 views

### How to find number of times sequence element $1$ is approached and from where?

Consider a sequence $1,-1,-1,-1,-1,-1,...,-1$. Start at the first element and move down the sequence according to the following rules: 1) If you jump from a $-1$ to another $-1$, turn the latter into ...
753 views

### IX-NAY on the IX-SAY

Will this sequence ever have a 6 in it? 9, 1, 1, 1, 10, 3, 1, 1, 10, 5, 1, 1, 10, 1, 5, 2, 1, 1, 10, 1, 1, 1, 5, 4, 1, 1, 10, 3, 1, 1, 5, 1, 1, 1, 5, 2, 1, 1, 10, 5, 1, 1, 5, 3, 1, 1, 5, 4, 1, ... ...
218 views

### I'm a Proper Divisor

Ann and Bob are playing a number game. Ann starts with the number 60. Then she subtracts a proper divisor of 60 from it. Bob then takes the number Ann made and subtracts one of its proper divisors ...
546 views

The sixteen players of a football squad, wearing shirts numbered 1 to 16, have arrived in town for a tournament. At their hotel, they are assigned 16 rooms consecutively numbered. Moreover, each of ...
616 views

### The Puzzling Reverse and Add Sequence

The sequence of numbers 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 11, 22,... (A056964 in the OEIS), in which the nth term equals n+reversal of digits of n, poses a number of intriguing puzzles. Here just ...
75 views

### Invert the numbers [duplicate]

I have an array of numbers containing 0 and 1 only and you are given a a constant C . You have to invert all the 0's to 1's by taking exactly C number of elements . What is the max number of 1's we ...
208 views

### Frobenius coin problem variation

Suppose you are give $n$ currency notes from $k$ to $k+n$ i.e $k, k+1,k+2.....k+n$ Where $k,n>0$ It's asked the total number of denomination of money that can't be formed using any number of ...
60 views

### How many ordered pairs (a,b) satisfy a^2=b^3+1, where a and b are integers? [closed]

(A)2 (B)3 (C)4 (D)5 I got 2 pairs (0,-1), and (3,2), but the correct answer is 5. Can somebody help? Thanks in advance!
87 views

### If $a$ and $b$ are two odd distinct prime number and if $a>b$, then $a^2-b^2$ can never be divided by [closed]

(A) 13 (B) 11 (C) 17 (D) None of the above
459 views

### Let M and N be single-digit integers. If the product 2M5 x 13N is divisible by 36, how many ordered pairs (M,N) are possible? [closed]

Let M and N be single-digit integers. If the product 2M5 x 13N is divisible by 36, how many ordered pairs (M,N) are possible? -- source I tried it by reducing 36 into its positive factors (1,2,3,...
1k views

### A king was born in a year that was a perfect square, lived a perfect square number of years, and also died in a year that was a perfect square

In which year could he have been born? (A) 1936 (B) 1764 (C) 1600 (D) 1444 The answer's (C). Why?
879 views

### Why did this prime-sequence puzzle not work?

While attacking a recent puzzle (whose solution ended up being entirely different from what I was trying), I was inspired to create a number-sequence puzzle with a sequence $(p_n)$ of primes where the ...
391 views

### Number sequence puzzle; 2, 10, 44 (+2 hints)

Predict the next three members of the sequence below and explain what the relationship is. 2, 10, 44, 1012, 248, This number sequence does not appear in the Online Encyclopedia of Integer ...
139 views

### Find the immediate square dancing neighbors, they dance together to perfect square

We live in a community of houses sequentially numbered from 1 to 100. We all love square dancing but only two immediate neighbors are joy to watch. If you concatenate their house numbers, it forms a ...
2k views

### We are two immediate neighbors who forged our own powers to form concatenated relationship. Who are we?

Our concatenated number is $\overline{ABAC},$ where $A, B, C$ are all positive digits (1 - 9). Our relationship is $$\overline{ABAC} = A^A + B^B + A^A + C^C$$ Who are we?
208 views

### SafeCracker #2 - The Mission Continues

Thanks to an alert StackE user, we were able to get the first safe open. Mission Details This next safe is in a former employee's house. He is gone for now, so we have no time to spare. We weren't ...
647 views

### Double or Take game

Double or Take is a two-player number game. Alice starts by selecting any positive integer. Bob's options are to: subtract a positive perfect square subtract a positive perfect cube, or double the ...
1k views

### Father and Son and Grandsons

The sum of the ages of a father, his son, and his two grandsons is 181. The father´s age has a common divisor greater than 1 with the ages of each of his three descendants, but not two of the latter ...
331 views

### Swap — A Puzzle I Created

This puzzle is called Swap. Let's find out why! Suppose you are given a random $\rm N\times N$ matrix (grid) with all the integers from $1$ to $\rm N^2$ each belonging in every grid square (a.k.a. ...
234 views

### What number + 1 equals itself? [closed]

Answer in numerical form. The answer is not an integer. The answer is in string form. The answer is not my love life...
1k views

### An Accountant Seeks the Help of a Mathematician

The accountant complaints to the mathematician: “I lent money to five other faculty members and still haven’t been paid back. You are one of them; the other four owe me 12 dollars altogether, but ...
862 views

### Consecutive integers around a circle

Find a block of positive consecutive integers that can be placed around a circle in some order so that any two adjacent numbers always have a common divisor greater than 1.
660 views

### The Legend of Four

As far as I know, all numbers have a root of 4. What I mean by this is as follows: Starting with any number, for example 384, I take the number of letters in that number. Then I repeat this process ...
728 views

### Digit sums of successive integers

For a natural number $x$ both, the digit sum of $x$ and the digit sum of $x+1$ are multiples of $7$. What is the smallest possible $x$?
369 views

### Can the sum, difference and product of 2 numbers be perfect squares? [closed]

If we take 2 numbers $x$ and $y$ such that $x>y>0$ and , can $x + y$, $x - y$ and $xy$ all be perfect squares?
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### Can you have the harmonic, geometric, arithmetic and quadratic mean of 2 numbers all being integers? [closed]

I created this problem looking at the hm-gm-am-qm inequality (hm = $\frac{2xy}{x + y}$, gm = $\sqrt{xy}$, am = $\frac{x + y}{2}$, qm = $\sqrt{\frac{x^2 + y^2}{2}}$ and hm $\le$ gm $\le$ am $\le$ qm). ...
5k views

### Create the numbers 1 - 30 using the digits 2, 0, 1, 9 in this particular order!

Inspired by the last year's "2018 four 4s challenge", I thought it's time to welcome 2019 by a similar challenge. This time you have to use the digits 2, 0, 1, 9 in this particular order to create the ...
535 views

### Finni's tricky game

Finni’s game: Person A thinks of a number (1 to 10). This number is called n. Person B says a number (1 to 10). This number is called x. Person A tells the absolute difference of n and x. This ...