![\"Puzzle](\"\/user\/pages\/02.blog\/03.problems\/p066-regex-crossword\/_puzzle.webp\" "\\"The")
At EuroPython 2024<\/a> I watched a lightning talk about the art of puzzle solving.\nIn it, the speaker showed the regex crossword puzzle that you can see above, that I took from this URL<\/a>.\nYou are supposed to fill the hexagonal grid with letters so that all of the regular expressions shown match.<\/p>\n If you don't know regular expressions, the puzzle only uses a simple subset of the syntax:<\/p>\n You can look this up and then you will be able to solve the challenge.\nYou can also use the site regex101<\/a> to help you check what each regular expression means.<\/p>\n Give it some thought!<\/p>\n<\/div>\n In the spirit of the puzzle hunting community, I will not share my solved grid here.\nIf you need help, feel free to email me<\/a> and we can talk it over.<\/p>\n P.S. if I understood correctly, in the puzzle hunting community you're not supposed to just fill in the grid.\nIn some way, somehow, you are supposed to be able to extract an English word or phrase from that filled puzzle without any extra information, and then you check that you got it correctly by \u201cchecking your answer spoiler-free\u201d in the website where I took the regex crossword from<\/a>\nI am still stuck in that step!<\/p>","summary":"Can you solve this crossword where all hints are regular expressions?","date_modified":"2025-07-23T16:49:02+02:00","tags":["mathematics","regex"],"image":"\/user\/pages\/02.blog\/03.problems\/p066-regex-crossword\/thumbnail.webp"},{"title":"Problem #065 \u2013 clock synchronisation","date_published":"2024-01-15T00:00:00+01:00","id":"https:\/\/mathspp.com\/blog\/problems\/clock-synchronisation","url":"https:\/\/mathspp.com\/blog\/problems\/clock-synchronisation","content_html":" When will these two clocks synchronise again?<\/p>\n\n You have two digital clocks.\nOne of them just displays the hours and minutes in the format HH:MM while the other is a stopwatch that displays minutes and seconds in the format MM:SS.\nThe stopwatch wraps around at 59:59, when it goes back to 00:00.<\/p>\n On a certain day at 00:00, the stopwatch is turned on.\nWhen will the clock and the stopwatch have matching displays again?\nAnd how often does that happen?<\/p>\n Give it some thought!<\/p>\n<\/div>\n The two clocks will have matching displays again at 01:01:01, then again at 02:02:02, 03:03:03, 04:04:04, ..., 23:23:23, so this happens 24 times per day.<\/p>\n To get to this conclusion, we start by realising that the stopwatch can actually be seen as a clock that shows the current time, but without the hours.\nThus, the minutes display of the regular clock will always match the minutes display of the stopwatch.<\/p>\n To recap, the two clocks have displays in the format HH:MM and MM:SS and we already know that the MM sections always match.\nThis means that for the two displays to display the same thing, we must have HH = MM = SS, which happens at 00:00:00, 01:01:01, ..., up to 23:23:23.<\/p>\n What if the stopwatch wraps around at 99:59 instead of 59:59?<\/p>","summary":"When will these two clocks synchronise again?","date_modified":"2025-07-23T16:49:02+02:00","tags":["mathematics"],"image":"\/user\/pages\/02.blog\/03.problems\/p065-clock-synchronisation\/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 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 Give it some thought!<\/p>\n<\/div>\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 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 Give it some thought!<\/p>\n<\/div>\n Remember<\/strong>:<\/p>\n Congratulations 🎉 to everyone who managed to solve this problem:\nCongratulations to you if you managed to solve this problem correctly!\nIf you did, feel free to<\/p>\n If you<\/em> managed to solve this problem, you can add your name to the list<\/a>!\nYou can also email me<\/a> your solution and we can discuss it.<\/p>\n A thing I like about this problem is that not only can you prove that interesting statement about the prime numbers, but you can also determine exactly what the lower bound \\(n_0\\)<\/span> is.<\/p>\n Let us say that \\(p_k\\)<\/span> is the \\(k\\)<\/span>-th prime.\nThen, if we set \\(n_0 = p_k\\)<\/span>, we are good to go.\nLet me show you why.<\/p>\n Suppose that \\(n\\)<\/span> is any integer \\(n \\geq n_0\\)<\/span>.\nThen, the interval \\([1, n]\\)<\/span> contains \\(p_k\\)<\/span> in it.\nWhy?\nBecause \\(p_k = n_0\\)<\/span> and \\(n \\geq n_0\\)<\/span>.<\/p>\n So, the interval \\([1, n]\\)<\/span> contains \\(k\\)<\/span> or more<\/em> prime numbers.\nIf it contains \\(k\\)<\/span> prime numbers, we just found our interval of length \\(n\\)<\/span> that contains exactly \\(k\\)<\/span> primes.\nIf it contains more than \\(k\\)<\/span> primes, we must do something about it.<\/p>\n If the interval \\([1, n]\\)<\/span> contains more than \\(k\\)<\/span> prime numbers, then we start sliding the interval to the right, like so:...<\/p>","summary":"Can you prove that there are arbitrarily many primes in arbitrarily big intervals? In this mathematical problem, which is just a logic challenge, the solution is surprisingly uncomplicated.","date_modified":"2025-07-23T16:49:02+02:00","tags":["logic","mathematics","number theory","primes"],"image":"\/user\/pages\/02.blog\/03.problems\/p063-arbitrarily-many-primes-in-arbitrarily-big-intervals\/thumbnail.webp"},{"title":"Problem #062 \u2013 sliding coins","date_published":"2022-06-12T00:00:00+02:00","id":"https:\/\/mathspp.com\/blog\/problems\/sliding-coins","url":"https:\/\/mathspp.com\/blog\/problems\/sliding-coins","content_html":" Can you align all of the coins on the right edge of the board?<\/p>\n\n The diagram above shows a 3 by 4 grid with 3 coins, one per row and one per column,\nsuch that the rightmost column of the grid is empty.<\/p>\n You are allowed to slide the coins to the columns on their right,\nbut the coins can never leave the board (nor can they be moved back to the left).\nAlso, you always have to move two coins at a time.<\/p>\n Thus, this is a legal move:<\/p>\n But moving a single coin, such as in the diagram below, is not allowed:<\/p>\n Your objective is to reach this position:<\/p>\n Can you do it?<\/p>\n What if you extend the board two columns to the left and two rows up,\nand add another two coins along the diagonal?<\/p>\n Give it some thought!<\/p>\n<\/div>\n This is an adaptation of a problem from the book “Algorithmic Puzzles”<\/a> by Anany Levitin and Maria Levitin.1<\/a><\/sup><\/p>\n Congratulations to the ones that solved this problem correctly and, in particular, to the ones\nwho sent me their correct solutions:<\/p>\n Know how to solve this?\nJoin the list of solvers by emailing me<\/a> your solution!<\/p>\n You can move all coins to the right in 3 moves:<\/p>\n This diagram shows the evolution of the grid:<\/p>\n If we extend the board up and left, we'd have this starting configuration:<\/p>\n Now, consider the total number of spaces to the right of the coins:<\/p>\n We see that there are 15 spaces on the grid to the right of coins.\nWe also know that each time we move two coins, the number of spaces to the right of the coins decreases by 2.\nNow, if we were to align all coins on the rightmost column, the number of spaces to the right of the coins would be 0.\nHowever, we can't...<\/p>","summary":"Can you align all of the coins on the right edge of the board?","date_modified":"2025-07-23T16:49:02+02:00","tags":["invariants","logic","mathematics"],"image":"\/user\/pages\/02.blog\/03.problems\/p062-sliding-coins\/thumbnail.webp"},{"title":"Problem #061 \u2013 flower garden","date_published":"2022-05-22T00:00:00+02:00","id":"https:\/\/mathspp.com\/blog\/problems\/flower-garden","url":"https:\/\/mathspp.com\/blog\/problems\/flower-garden","content_html":" Three mathematicians discuss a beautiful flower garden and the coloured flowers within.<\/p>\n\n Three mathematicians were discussing a beautiful flower garden.\nThe first mathematician said:<\/p>\n \u201cIt's a beautiful garden.\nAll flowers are either red, yellow, or blue...\nAnd whatever three flowers you pick, one of them is bound to be red.\u201d<\/p>\n<\/blockquote>\n The second mathematician said:<\/p>\n \u201cIt is indeed a beautiful garden!\nThere are red flowers, yellow flowers, and blue flowers!\nI also noticed that, whatever three flowers you pick, one of them is bound to be yellow.\u201d<\/p>\n<\/blockquote>\n The third mathematician, who had never been to the garden, remarked:<\/p>\n \u201cThat's interesting!\nIf that's true, I can say that, whatever three flowers I pick, one of them is bound to be blue!\u201d<\/p>\n<\/blockquote>\n The first two mathematicians glanced at each other and smiled.<\/p>\n Is the third mathematician right? Or wrong? Why?<\/p>\n Give it some thought!<\/p>\n<\/div>\n This problem is from the amazing [book \u201cTo Mock a Mockingbird\u201d][mock-mockingbird] by the mathematician Raymond Smullyan.1<\/a><\/sup><\/p>\n Congratulations to the ones that solved this problem correctly and, in particular, to the ones\nwho sent me their correct solutions:<\/p>\n Know how to solve this?\nJoin the list of solvers by emailing me<\/a> your solution!<\/p>\n The third mathematician is right.<\/p>\n The first sentence of each mathematician establishes that all flowers in the garden are one of three colours (red, yellow, blue) and that all colours are represented; that is, there is at least<\/em> one red flower, at least<\/em> one yellow flower, and at least<\/em> one blue flower.<\/p>\n Then, the first mathematician says that whatever three flowers you pick, one is bound to be red.\nThis means that the garden cannot contain three or more flowers that are not red.\nAfter all, if there were, say, two yellow flowers and one blue flower,\nyou could pick those three and there wouldn't be a red flower in there.<\/p>\n At the same time, we know there has to be at least one yellow flower and one blue flower,\nso we conclude the garden has an unknown number of red flowers, one yellow flower, and one blue flower.<\/p>\n Then, the second mathematician speaks!\nFrom their remark, we could make a similar independent conclusion:\nthe garden has an unknown number of yellow flowers, one red flower, and one blue flower.<\/p>\n Thus, if we combine the two conclusions,\nwe get that the garden has three flowers only:<\/p>\n Therefore, the third mathematician is also right:\nwhatever three flowers you pick, one is bound to be blue.<\/p>\n Don't forget to subscribe to the newsletter<\/a> to get bi-weekly\nproblems sent straight to your inbox.<\/p>\n This is an Amazon Affiliate link and I may earn a commission if you purchase the book. This comes at no extra cost to you.\u00a0\u21a9<\/a><\/p>\n<\/li>\n<\/ol>\n<\/div>","summary":"Three mathematicians discuss a beautiful flower garden and the coloured flowers within.","date_modified":"2025-07-23T16:49:02+02:00","tags":["logic","mathematics"],"image":"\/user\/pages\/02.blog\/03.problems\/p061-flower-garden\/thumbnail.png"},{"title":"Problem #060 \u2013 realtor commissions","date_published":"2022-05-08T00:00:00+02:00","id":"https:\/\/mathspp.com\/blog\/problems\/realtor-commissions","url":"https:\/\/mathspp.com\/blog\/problems\/realtor-commissions","content_html":" Two realtors discuss who's netting the award for highest average commission, but it isn't clear who the winner is...<\/p>\n\n Alice and Bob are two realtors that work at the same premium real estate agency, WSH.\nBecause WSH really wants to focus on selling premium properties,\nthey instated a monetary prize that is awarded every two years to the realtor that has the highest average commission over the past two years.<\/p>\n The award for the years 2020 and 2021 is being attributed soon.\nOne day, at the office, Alice overheard Bob gloating:<\/p>\n “I can't wait to get my money! I already know what I'll be doing with it!”<\/p>\n<\/blockquote>\n Alice turned to Bob and asked:<\/p>\n “How can you be so sure that you are the one netting the prize?”<\/p>\n<\/blockquote>\n Bob smirked and returned:<\/p>\n “I guess you haven't been paying attention, Alice!\nIn 2020 I was the realtor with the highest average commission over that year...\nAnd in 2021 I did it again!”<\/p>\n<\/blockquote>\n Alice smiled and returned to work.<\/p>\n Who do you think is right?\nIs Bob definitely getting the award?\nOr is there a scenario in which Alice gets the prize?<\/p>\n Give it some thought!<\/p>\n<\/div>\n Congratulations to the ones that solved this problem correctly and, in particular, to the ones\nwho sent me their correct solutions:<\/p>\n Know how to solve this?\nJoin the list of solvers by emailing me<\/a> your solution!<\/p>\n\n
.<\/code>;<\/li>\n|<\/code>;<\/li>\n?<\/code>, +<\/code>, and *<\/code>;<\/li>\n[...]<\/code> and negated character sets with [^...]<\/code>; and<\/li>\n()<\/code> and group references with \\1<\/code>, \\2<\/code>, etc.<\/li>\n<\/ul>\nSolution<\/a><\/h2>\n
Problem statement<\/a><\/h2>\n
Solution<\/a><\/h2>\n
Bonus question<\/a><\/h3>\n
Problem statement<\/a><\/h2>\n
Solution<\/a><\/h2>\n
Problem statement<\/a><\/h2>\n
Solvers<\/a><\/h2>\n
Solution<\/a><\/h2>\n
![\"\"](\"\/user\/pages\/02.blog\/03.problems\/p062-sliding-coins\/_setup.webp\" "\\"Diagram")
Problem statement<\/a><\/h2>\n
![\"The diagram showing an example of a legal move with some arrows.\"](\"\/user\/pages\/02.blog\/03.problems\/p062-sliding-coins\/_legal_move.webp\" "\\"A")
![\"The diagram showing an example of an illegal move.\"](\"\/user\/pages\/02.blog\/03.problems\/p062-sliding-coins\/_illegal_move.webp\" "\\"An")
![\"The same 3 by 4 grid containing a coin per row, with all coins on the rightmost column.\"](\"\/user\/pages\/02.blog\/03.problems\/p062-sliding-coins\/_objective.webp\" "\\"Target")
Solvers<\/a><\/h2>\n
Solution<\/a><\/h2>\n
X . . . start\n. X . .\n. . X .\n\n. X . . >\n. X . .\n. . . X >\n\n. . X . >\n. . X . >\n. . . X\n\n. . . X >\n. . . X >\n. . . X<\/code><\/pre>\nX . . . . .\n. X . . . .\n. . X . . .\n. . . X . .\n. . . . X .<\/code><\/pre>\nX 1 2 3 4 5\n. X 6 7 8 9\n. . X 0 1 2\n. . . X 3 4\n. . . . X 5<\/code><\/pre>\n![\"\"](\"\/user\/pages\/02.blog\/03.problems\/p061-flower-garden\/thumbnail.png\" "\\"Photo")
Problem statement<\/a><\/h2>\n
\n
\n
\n
Solvers<\/a><\/h2>\n
\n
Solution<\/a><\/h2>\n
\n
\n\n
![\"\"](\"\/user\/pages\/02.blog\/03.problems\/p060-realtor-commissions\/thumbnail.png\" "\\"Photo")
Problem statement<\/a><\/h2>\n
\n
\n
\n
Solvers<\/a><\/h2>\n
Solution<\/a><\/h2>\n