# Pointless but interesting math puzzles **URL:** \ **Category:** Miscellaneous and Personal Stuff I Must Share\ **Created:** [July 24, 2016, 4:23am UTC](https://boards.straightdope.com/t/pointless-but-interesting-math-puzzles/760745 "2016-07-24T04:23:07Z")\ **Posts on this page:** 12\ **Page:** 2
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [July 29, 2016, 1:16am UTC](https://boards.straightdope.com/t/pointless-but-interesting-math-puzzles/760745/21 "2016-07-29T01:16:02Z")
My grandmother(\*) used to say: “I ask what trump is often — it shows I’m interested in the game!” - - Paternal grandmother. My maternal grandmother was an ACBL Life Master. > [@septimus](#): > > … the _a priori_ chances for these two cases are about the same. (1st case is slightly more likely than 2nd.) > [@Hari\_Seldon](#): > > Slight nitpick: The odds aren’t _quite_ 2/3 since a 2-2 distribution is slightly more likely than 3-1. ---
**Author:** ![Velocity](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/velocity/32/18006_2.png) [@Velocity](https://boards.straightdope.com/u/Velocity)\ **Post date:** [July 29, 2016, 1:23am UTC](https://boards.straightdope.com/t/pointless-but-interesting-math-puzzles/760745/22 "2016-07-29T01:23:03Z")
A puzzle I mused about yesterday: Using the current Electoral College map, what outcomes (other than 0-2 and 566-567 EVs) are mathematically _unattainable_? Can a candidate get 423 EVS? 17? 196? Etc. etc. ---
**Author:** ![Velocity](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/velocity/32/18006_2.png) [@Velocity](https://boards.straightdope.com/u/Velocity)\ **Post date:** [July 29, 2016, 1:33am UTC](https://boards.straightdope.com/t/pointless-but-interesting-math-puzzles/760745/23 "2016-07-29T01:33:58Z")
Ooops, I mean 536-537. ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [July 29, 2016, 1:35am UTC](https://boards.straightdope.com/t/pointless-but-interesting-math-puzzles/760745/24 "2016-07-29T01:35:46Z")
That’s equivalent to (one form of) the [Knapsack problem](https://en.wikipedia.org/wiki/Knapsack_problem), which is well-known to be a hard problem in a precise mathematical sense. That said, there are only 50 states, and so one could brute-force the answer pretty easily. The best algorithms require (I think) 2[sup]n/2[/sup] steps, which isn’t very much for a computer. Kinda tempting, actually… ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [July 29, 2016, 2:56am UTC](https://boards.straightdope.com/t/pointless-but-interesting-math-puzzles/760745/25 "2016-07-29T02:56:16Z")
Don’t forget that Maine and Nebraska award electoral votes by district: Nebraska might split its 5 votes 3-2, 4-1, or 5-0. Because of this I’m rather certain _any split of the 538 votes is possible._ I won’t provide a rigorous vote (although it would be easy to construct one) but the large number of states with only 3 or 4 votes (along with Nebraska and Maine ready to provide single votes as needed) means that any split is achieved easily. ---
**Author:** ![ITR\_champion](https://avatars.discourse-cdn.com/v4/letter/i/c67d28/32.png) [@ITR\_champion](https://boards.straightdope.com/u/ITR_champion)\ **Post date:** [July 29, 2016, 4:06am UTC](https://boards.straightdope.com/t/pointless-but-interesting-math-puzzles/760745/26 "2016-07-29T04:06:58Z")
On the remote Island of Gugu, the language of the islanders has a strange quirk. For any word, if you take any portion of the word and double it, the word still means the same thing. For for example: The means the same as Thhe computer means the same as computputer elephant means the same as elephanephant Show that kol means the same as kolokol. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 29, 2016, 4:12am UTC](https://boards.straightdope.com/t/pointless-but-interesting-math-puzzles/760745/27 "2016-07-29T04:12:14Z")
Ah, the word problem for an idempotent monoid. Well, kol = kokol (double ko) = kokolol (double ol) = kokolokolol (double okol, or, just as well, double kolo) = kolokol (un-double koko and un-double olol) ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [July 29, 2016, 4:18am UTC](https://boards.straightdope.com/t/pointless-but-interesting-math-puzzles/760745/28 "2016-07-29T04:18:34Z")
I like it![spoiler]kol =\> kolol and kolol =\> kolokolol But also: kolokol =\> kolokolol kolokolol = kolokolol, so they’re the same[/spoiler] (Ninjad by **Indistinguishable** , dagnabit. But I learned a new word.) ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [July 29, 2016, 5:53am UTC](https://boards.straightdope.com/t/pointless-but-interesting-math-puzzles/760745/29 "2016-07-29T05:53:38Z")
Speaking of identical adjacent substrings … here’s a famous old problem that may take much ingenuity to solve. For maximum kudos don’t be satisfied with the “usual” solution, findable via Google, but come up with a novel alternate approach! _Find an arbitrarily long string on the alphabet {a,b,c} such that no two adjacent substrings are identical._ For clarity, here’s a start which does NOT work: _abacaba…_ You’ve already “painted yourself into a corner” because you must append an a, b, or c but none of them works: abacab **a\*\*\*\*a** abac **ab\*\*\*\*ab** **abac\*\*\*\*abac** ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [July 29, 2016, 6:49pm UTC](https://boards.straightdope.com/t/pointless-but-interesting-math-puzzles/760745/30 "2016-07-29T18:49:04Z")
> [@septimus](#): > > _Find an arbitrarily long string on the alphabet {a,b,c} such that no two adjacent substrings are identical._ This may have been too difficult a problem for a casual thread. Just find a solution via Google or ignore the problem altogether. There may be little noteworthy about the alternate solution I alluded to — it just has sentimental value to me! That solution came to me in a flash 50 years ago when I was a boy; the other identical adjacent substring problem reminded me. 😛 _ **Or …** _ find a satisfactory such arbitrarily long string in which the symbols a,b,c do NOT occur with equal (1/3, 1/3, 1/3) rates. AFAIK the existence of such is unsolved. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 29, 2016, 8:15pm UTC](https://boards.straightdope.com/t/pointless-but-interesting-math-puzzles/760745/31 "2016-07-29T20:15:09Z")
Well, I know some standard solutions and therefore avoided saying anything. What’s the alternate solution you have in mind? ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [July 29, 2016, 8:39pm UTC](https://boards.straightdope.com/t/pointless-but-interesting-math-puzzles/760745/32 "2016-07-29T20:39:15Z")
Given a satisfactory sequence, we only need show a way to convert it into a longer satisfactory sequence, e.g. three times as long (or a little longer). Brief summary without rigorous proof follows. Step 1) Duplicate letters that separate identical strings. For example, replace _cbabcba_ with _cbabbcba_. Call this new string Z. (This string is in violation, but this is just an intermediate form; the key step follows.) Step 2) Construct a new string c\_\_c\_\_c\_\_c\_\_c\_\_c\_\_ where the gaps between c’s are filled with 1, 2 or 3 letters (a or b) according as the letters in Z are a, b, or c. (Except at the very beginning of the string, there will be just one compliant way to do this.) For example, if Z = _abca…_ (corresponding to 1,2,3,1) then the new string will be _cacbacabacbc…_. The gaps between c’s in the new string are of lengths 1,2,3,1. Half a century later, I recall swinging on our backyard swing when this happy solution popped fully formed into my head. I knew it worked … and could prove that fact rigorously with much greater ease than I could now. (Can this be adapted to get more than 33.33% c’s? I very dimly recall once randomly generating long compliant strings with well over 33.33% c’s, but very dim is the operative phrase here.) [Previous page](https://boards.straightdope.com/t/pointless-but-interesting-math-puzzles/760745.md?page=1)