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 a factor #1

The series of Find a factor puzzle is started by Culver Kwan, and asks the solver to identify a factor of a certain large number within a certain range using some mathematical identities. This should ...
9
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0answers
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Dead By Daylight

This puzzle is a reference to a game, Dead By Daylight. Though you don't need a knowledge about the game. Consequently, there is no "video-games" tag. Story The Entity wants to have the killers' ...
4
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1answer
161 views

What's the graph relation? #3

What's the relation that joins the nodes? Previous What's the graph relation? #1 What's the graph relation? #2
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2answers
371 views

Primes in a Line

Place the first 20 primes (2 to 71) in a line so that the sum or difference (or both) of any two primes that find themselves next to each other is always a perfect square. For which other values of N ...
6
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1answer
147 views

Follow the path of relation through the grid #3

There is a relation between rectilinear-adjacent squares such that there is a unique rectilinear path from the top-left corner of the grid down to the bottom-right corner of the grid. Each square can ...
7
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1answer
172 views

Follow the path of relation through the grid #2

There is a relation between rectilinear-adjacent squares such that there is a unique rectilinear path from the top-left corner of the grid down to the bottom-right corner of the grid. Each square can ...
9
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2answers
222 views

Add a divisor! A game

Let $k$ be a positive integer. Amy and Ben are playing a game, with the number $1$ written on the whiteboard initially. Amy and Ben do the following in order, starting with Amy: Suppose the ...
11
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1answer
291 views

Floor floor floor inside another floor

Inspired from this question $$ \aleph(x,n)=\lfloor x\lfloor x\lfloor x...\rfloor\rfloor\rfloor\ $$ where $\aleph$ is the inner floor function with $n$ times for $x$. For example; $$ \aleph(x,3)=\...
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3answers
176 views

The ceiling function and powers of 2

How many integers $1\le x\le2048$ such that $$\Big\lceil \frac x{2^n}\Big\rceil$$ is not a multiple of five for all nonnegative integers $n$? This problem is a 2020 contest problem which has finished....
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37 views

Brute force a keypad with minimal keystrokes [duplicate]

Senario Say you have a keypad whose password is some two digit code which you do not know, say 34. Entering digits in succession on this keypad eg. ...
45
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4answers
6k views

x⌊x⌊x⌊x⌋⌋⌋ = 2020

Solve for $x$: $$ x \left\lfloor x \left\lfloor x \left\lfloor x \right\rfloor \right\rfloor \right\rfloor = 2020. $$ The floor function $\left\lfloor t \right\rfloor$ has the usual “greatest integer ...
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2answers
378 views

Consecutive numbers which use all digits a different number of times

Are there arbitrarily long sets of consecutive numbers such that when writing the set down, every single digit (0 to 9) is used a different number of times?
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3answers
170 views

The death prism

One day, you are caught by a evil wizard. He presents you with a prism, and says, "You can ask me to turn this prism to any $n$-angled right prism. Then you shall fill in $1$ to $3n$ with no ...
6
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1answer
201 views

What's the graph relation? #2

What's the relation that joins the nodes? Open the image in a new tab if you'd like to see the diagram with better resolution. Previous What's the graph relation? #1 Hint 1
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1answer
92 views

What's the graph relation? #1

What is the relation that connects the nodes of this digraph?
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4answers
1k views

What is larger than largest, your intuitions are tarnished. (What am I?)

You might call me a number, but that would be a blunder. In class you may have been told otherwise, but I tell you now that those were all lies. I'm in the deck of the cards, as big as them come. ...
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0answers
286 views

I'm a number with a special product, so name me when you think you've got it

In the blocks that come before, their special product tells us more. To guess my scheme you'll need calculation, but only little tests of recreation. Each block contains atoms strong as Thor, ...
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2answers
980 views

First digit of 2020!

This is a follow up to First digit of 3^2020 Can you find the first digit of 2020! (factorial) without a computer?
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3answers
5k views

First digit of 3^2020

Inspired by The last digit for 3^(2019) Can you find the first digit of $3^{2020}$ without a computer?
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2answers
153 views

The Balls of Death

You have nightmares about a pool ball with the number 1 on it, and an empty box. Why? Immortality Imagine, if you will, we are 5000 years into the future. Medicine has evolved and we now are ...
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2answers
697 views

The last digit for 3^(2019)

Which would be the last digit for $3^{2019}$ ? You can And afterwards
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4answers
2k views

What's in my pocket?

Well, I can tell you Johnny has memory cards in his pocket. Back Story My brother, Johnny, is a tech nerd. He loves gadgets of all kinds. As a matter of fact, you can be sure at any one given time, ...
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3answers
432 views

Number Guessing (Part 1)

I thought up two positive integers with product less than $500$. I told their product to Penny, and their sum to Sandy, and told both of them the constraints and they are both perfect logicians. ...
5
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1answer
5k views

How to programmatically solve math puzzle

I have this puzzle and I want to solve by code. I wrote simple code trying to brute force but it fail in one condition https://dotnetfiddle.net/cJyu3w Anyone know of a link or source how to solve ...
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1answer
1k views

Four mathematicians and their ages

Four mathematicians, none yet a centenarian, meet for coffee. The graph-theorist among them noticed that the common divisor graph of their ages (that is, the graph whose vertices are their ages, two ...
4
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1answer
145 views

Playing Collatz with graphs

a) Given a set of positive integers, its common divisor graph (CD-graph) is the graph whose vertices are the integers, two of which are joined by an edge if (and only if) they have a common divisor ...
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5answers
2k views

Nice no-computers way to find limerick primes?

A limerick number is a 5-digit number whose digits are in the form of a limerick rhyme scheme: $aabba$. How many limerick primes (limerick numbers which are also prime) are there? The answer is ...
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4answers
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Professor Halfbrain and the number cycles

Yesterday afternoon I met professor Halfbrain at an art gallery. The professor looked tired and exhausted. He told me that he had spent many working days and many sleepless nights with lengthy ...
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1answer
208 views

Puzzling set of numbers [closed]

Let's have the following numbers $23, 40, 42, 44, \sqrt{43},\ 128i, 130, \sqrt{172}\ $. What is the relationship between these numbers, taken four at a time? There are only two combinations when you ...
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6answers
643 views

A special triple of factors

Using each of the digits 1,2,3,4,5,6,7,8,9 exactly once, create three 3-digit numbers such that their product is a maximum.
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1answer
172 views

Generating special numbers [closed]

Here are three numbers that are related to one another: 9841, 8591, 4800 How are these numbers related and how are they generated?
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4answers
909 views

Making the whole set into primes

Let's say you start with a set of sequential integers starting from 2, so: $ 2, 3, 4, 5, \dots, N $ for some $ N > 2. $ The goal is to use identical basic arithmetic operations ($ +, -, \times, \...
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3answers
202 views

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 ...
8
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3answers
640 views

Discover the six-character password!

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 ...
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0answers
81 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
437 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, ...
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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
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4answers
723 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
750 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
564 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
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2answers
585 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
596 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
247 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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3answers
1k 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
148 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
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2answers
130 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
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1answer
177 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
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2answers
193 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
757 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, ... ...

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