{"version":"https:\/\/jsonfeed.org\/version\/1","title":"mathspp.com feed","home_page_url":"https:\/\/mathspp.com\/blog\/tags\/logic","feed_url":"https:\/\/mathspp.com\/blog\/tags\/logic.json","description":"Stay up-to-date with the articles on mathematics and programming that get published to mathspp.com.","author":{"name":"Rodrigo Gir\u00e3o Serr\u00e3o"},"items":[{"title":"N queens problem","date_published":"2024-02-09T00:00:00+01:00","id":"https:\/\/mathspp.com\/blog\/n-queens-problem","url":"https:\/\/mathspp.com\/blog\/n-queens-problem","content_html":"

This article shows how to solve the N queens problem in 20 lines of code.<\/p>\n\n

The problem for N = 10<\/a><\/h2>\n

(You can skip to the solution<\/a> or just see the code<\/a>.)<\/p>\n

“Do you understand what you have to do?”, Queen #6 asks.<\/p>\n

“Uh, I– I think so.”<\/p>\n

The challenge sounded simple enough but I was a bit intimidated by the Ten Queens all looking at me.<\/p>\n

They stood tall in front of me.\nAn impeccable pose, worthy of a Queen.\nTimes ten.<\/p>\n

They looked rigid and cold, wearing their white and black dresses.\nBut if you looked carefully enough, they also looked...\nThey looked... hopeful!\nThey believed I would be able to help them.<\/p>\n

I wasn’t feeling confident and Queen #8 picked up on that, so she decided to recap what they needed:\n“Like my sister #6 said, we need to distribute ourselves on this 10 × 10 board.\nThe goal is to count<\/em> in how many different ways this can be done.<\/p>\n

We like to have room to pace a bit, so no two of us can be on the same row, column, or diagonal.\nThis restriction is essential.”<\/p>\n

I nodded along while she recapped.\nThen, I asked “How on Earth am I supposed to compute this?\nI can’t do this with pen and paper!”.<\/p>\n

All Queens started laughing uncontrollably.\nQueen #1 managed to control herself for long enough to reply.<\/p>\n

”You silly! You use Python, of course!”\nShe waved.<\/p>\n

The two pawns at the entrance of the room we were in opened the massive doors.\nAs the huge doors creaked and opened slowly, four pawns came into the hall, carrying a computer.\nThe computer was already turned on.\nAs the computer moved closer, this is what I saw on the screen:<\/p>\n

Python 3.12.0 (the Ten Queens build)\nType \"help\" for more information.\n>>><\/code><\/pre>\n
“You have one week.\nYou can start now.”<\/p>\n
I walked up to the computer and started working on the problem.<\/p>\n
The solution<\/a><\/h2>\n
I was thinking aloud while I was typing.<\/p>\n
“We know that no two queens can be on the same row or column.\nAnd that's easy to enforce in my code.\nI'll traverse the columns and put one of you in each column while also not repeating rows.\nIt's the diagonals that I have to be careful about.”<\/p>\n
I paused for a bit.\nThen I wrote this function:<\/p>\n
def diagonally_safe(row, col, placements):\n    for qrow, qcol in enumerate(placements):\n        if row - col == qrow - qcol or row + col == qrow + qcol:\n            return False\n    return True<\/code><\/pre>\n
I proceeded to explain:<\/p>\n
“I'll store a tuple called placements<\/code> with the positions of some of you.\nThe index is the row you're in and the value itself represents the column.\nFor example, if placements<\/code> is (5, 0, 4)<\/code> that means that the row 0<\/code> has a queen in column 5<\/code>, the row 1<\/code> has a queen in column 0<\/code>, and the row 3<\/code> has a queen in column 4<\/code>.”<\/p>\n
The queens...<\/p>","summary":"This article shows how to solve the N queens problem in 20 lines of code.","date_modified":"2025-09-27T16:50:04+02:00","tags":["algorithms","combinatorics","logic","programming","python","recursion"],"image":"\/user\/pages\/02.blog\/n-queens-problem\/thumbnail.webp"},{"title":"Problem #064 \u2013 infinite mathematicians with hats","date_published":"2024-01-02T00:00:00+01:00","id":"https:\/\/mathspp.com\/blog\/problems\/infinite-mathematicians-with-hats","url":"https:\/\/mathspp.com\/blog\/problems\/infinite-mathematicians-with-hats","content_html":"
How can an infinite number of mathematicians figure out their own hat colours?<\/p>\n\n
Problem statement<\/a><\/h2>\n
An infinite number of mathematicians are standing in a line.\nIn a couple of minutes, a disruption in the space-time continuum will cause a black or white hat to appear on the head of each mathematician.\nAfter that happens, the mathematicians will try to guess their own hat colour (they can't see it) based on the colours of the hats of all other mathematicians.<\/p>\n
What's the strategy that the mathematicians must agree on, before hand, so that only a finite<\/em> number of mathematicians guesses wrong?<\/p>\n
\n
Give it some thought!<\/p>\n<\/div>\n
Solution<\/a><\/h2>\n
When we look at each matematician's hat we can create a sequence of \\(0\\)<\/span>s and \\(1\\)<\/span>s.\nFor example, if everyone has a white hat except for the first three mathematicians, then we would have the sequence \\((1,1,1,0,0,\\cdots)\\)<\/span>.<\/p>\n
Let us say two sequences are alike<\/em> if they are the same except for a finite number of positions.\nAs an example, the sequence we get when all mathematicians have a white hat and the previous sequence are alike<\/em> because they differ only in the first three positions: \\((\\underline{0}, \\underline{0}, \\underline{0}, 0, 0,\\cdots)\\)<\/span>.<\/p>\n
For any sequence \\(S\\)<\/span> we can think of all the other sequences with which \\(S\\)<\/span> is alike<\/em>.\nLet us call that group of alike<\/em> sequences \\([S]\\)<\/span>.\nWhat the mathematicians must do is: for every single group \\([S]\\)<\/span> they must pick a random sequence from that group and memorise it.<\/p>\n
When they are given the hats, they must look at their colleagues and figure out in which group \\([S]\\)<\/span> their hat sequence belongs.\nThen, they will recall the sequence they all memorized from that group and each mathematician will say that their hat is of the colour corresponding to their position on the memorized sequence.\nBy definition, only a finite number of mathematicians guesses wrong because the sequence they will recreate and the real sequence are alike<\/em>, i.e. they differ only in a finite number of positions.<\/p>","summary":"How can an infinite number of mathematicians figure out their own hat colours?","date_modified":"2025-07-23T16:49:02+02:00","tags":["logic","mathematics","sequences"],"image":"\/user\/pages\/02.blog\/03.problems\/p064-infinite-mathematicians-with-hats\/thumbnail.webp"},{"title":"Problem #063 \u2013 arbitrarily many primes in arbitrarily big intervals","date_published":"2023-01-29T00:00:00+01:00","id":"https:\/\/mathspp.com\/blog\/problems\/arbitrarily-many-primes-in-arbitrarily-big-intervals","url":"https:\/\/mathspp.com\/blog\/problems\/arbitrarily-many-primes-in-arbitrarily-big-intervals","content_html":"
Can you prove that there are arbitrarily many primes in arbitrarily big intervals?<\/p>\n\n
Problem statement<\/a><\/h2>\n
The problem statement asks you to prove that for any<\/em> positive integer number \\(k\\)<\/span> you can think of,\nthere will be a certain lower bound \\(n_0\\)<\/span> (that depends on \\(k\\)<\/span>) such that for any<\/em> integer \\(n \\geq n_0\\)<\/span>,\nthere is<\/em> an interval of length \\(n\\)<\/span> that contains exactly<\/em> \\(k\\)<\/span> primes.\n(When we talk about the length of the interval, we are talking about how many integers it contains.)<\/p>\n
If the statement confused you a bit, that is ok.\nLet me rephrase it.<\/p>\n
You and I will play a game.\nI will think of a positive integer \\(k\\)<\/span>.\nNow, your job is to come up with another positive integer \\(n_0\\)<\/span> such that,\nif I pick a number \\(n\\)<\/span> greater than \\(n_0\\)<\/span>,\nyou can always find an interval of size \\(n\\)<\/span> that contains \\(k\\)<\/span> primes.<\/p>\n
For example, if I thought of \\(k = 5\\)<\/span>, you could not<\/em> pick \\(n_0 = 4\\)<\/span>.\nWhy not?\nBecause if I pick \\(n = 4\\)<\/span>, there is no<\/em> interval of length \\(4\\)<\/span> that contains \\(5\\)<\/span> prime numbers...\nEspecially because an interval of length \\(4\\)<\/span> contains only \\(4\\)<\/span> integers!<\/p>\n
This problem was posed to me by my mathematician cousin and I confess that worried me a bit.\nFunnily enough, the problem has a surprisingly simple solution.\n(I am not saying you will get there easily.\nI am just saying that once you do, you will realise the solution was not very complicated.)<\/p>\n
\n
Give it some thought!<\/p>\n<\/div>\n
Remember<\/strong>:<\/p>\n