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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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-4
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0answers
100 views

Express $1-2-3-4…$ using one summation only ($\sum$), in the simplest way possible

No taking outside any constant/variable, and also the goal is to make it simple as possible.
0
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0answers
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 ...
7
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2answers
412 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, ...
17
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4answers
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 ...
10
votes
4answers
670 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 ...
9
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3answers
737 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 (...
8
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1answer
544 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: ...
10
votes
2answers
537 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 ...
10
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2answers
585 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 ...
11
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4answers
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.
7
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0answers
210 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 ...
16
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1answer
889 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 ...
6
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2answers
142 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 ...
5
votes
2answers
115 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 ...
6
votes
1answer
170 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 ...
2
votes
2answers
177 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 ...
8
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2answers
746 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, ... ...
6
votes
1answer
213 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 ...
9
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4answers
543 views

The Football Squad

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 ...
12
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2answers
607 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 ...
0
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0answers
73 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 ...
1
vote
1answer
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 ...
1
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0answers
58 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!
-1
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2answers
83 views
3
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2answers
447 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,...
4
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2answers
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?
5
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4answers
873 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 ...
8
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1answer
375 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 ...
6
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1answer
138 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 ...
17
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2answers
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?
5
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2answers
203 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 ...
8
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3answers
643 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 ...
6
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2answers
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 ...
6
votes
2answers
327 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. ...
-7
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2answers
227 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...
7
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2answers
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 ...
20
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4answers
812 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.
8
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5answers
653 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 ...
13
votes
2answers
720 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 $x$?
2
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2answers
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?
3
votes
2answers
89 views

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). ...
11
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4answers
4k 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 ...
4
votes
8answers
532 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 ...
3
votes
1answer
159 views

Three-digit multiplication puzzle, part III: Return of the Hex

Followup to: Three-digit multiplication puzzle and Three-digit multiplication puzzle, part II: ever heard of senary? Place different three-digit hexadecimal numbers (000-FFF) on each of the seven ...
3
votes
2answers
152 views

Three-digit multiplication puzzle, part II: ever heard of senary?

Followup to: Three-digit multiplication puzzle I'd never heard of "senary" either, until creating this series of three puzzles. It seems that "senary" is the word for base-six notation. Who knew? ...
4
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1answer
190 views

Three-digit multiplication puzzle

Place different three-digit numbers (000-999) on each of the seven nodes of the following diagram: There are six lines in the diagram, on each of which are three of the nodes. On each of those lines, ...
7
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1answer
512 views

Fourteen numbers around a circle

Place the numbers 1 to 14 around this circle so that both the sum and (absolute) difference of any two neighboring numbers is a prime.
6
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2answers
328 views

The Magic Letter H

Place seven different positive integers on the empty disks of the H figure below so that the product of the three numbers in any straight black line is always the same. Now place seven other numbers ...
14
votes
1answer
750 views

My Social Security Card Number

I have forgotten my social security card number. All I remember is that it is the largest integer with the property that the block of any two of its digits that are adjacent is either a two-digit ...
9
votes
2answers
1k views

Ten-digit number that satisfy divisibilty rules for 2,3,4,5,6,7,8,9,10&11

Question: Arrange the digits 1,2,3,4,5,6,7,8,9,0 to make a ten-digit Number that satisfies all of the divisibility rules for 2,3,4,5,6,8,9,10,&11. BONUS: make the number also divisible by 7