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

\n# Solution<\/a><\/h1>\n## Bonus question<\/a><\/h2>\n# Problem statement<\/a><\/h1>\n*\n*# Solution<\/a><\/h1>\n*Problem statement**<\/a><\/h1>\n**\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":"2024-01-15T17:33:24+01:00","tags":["mathematics"],"image":"\/user\/pages\/02.blog\/02.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":"2024-01-15T17:33:24+01:00","tags":["logic","mathematics","sequences"],"image":"\/user\/pages\/02.blog\/02.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**

**there are infinitely many primes; however<\/li>\n****they become scarcer and scarcer the further you go down the number line.<\/li>\n<\/ul>**# Solvers<\/a><\/h1>\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

- Rodrigo G. S., Portugal 🇵🇹 (<- example);<\/li>\n<\/ul>
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**Solution<\/a><\/h1>\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":"2024-01-15T17:33:24+01:00","tags":["logic","mathematics","number theory","primes"],"image":"\/user\/pages\/02.blog\/02.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*

- Rodrigo G. S., Portugal 🇵🇹 (<- example);<\/li>\n<\/ul>