Another lightbulb puzzle

So last week was the first weekly Joburg Math meetup. Which basically means a bunch of friends meeting up and playing with math olympiad type problems.  If anyone is interested these meetups happen on  Sundays at noon in Motherland coffee Rosebank Mall, and of course everyone is welcome. hopefully they'll also give me new ideas for more regular math type blog posts.

Anyway here is a problem from The book Creative Mathematics by Alan Beardon, and soem generalizaions due to those involved in the meetup.

Problem 1. I have a line of n lightbulbs. Each lightbulb has a switch associated to it. The switch toggles (if on changes to off, if off changes to on) all bulbs except the one it's associated to.  If all the bulbs start out as off, for whcih n can they be changed to all on?

Problem 2. What if I put the bulbs in an n1 by n2 grid. Again each bulb has a switch associated to it. The switch toggles all bulbs sharign either a row or coloumn with it's bulb.  Another way to say that is that it toggles bulbs agreeing in with it's position in exactly one coordinate.

Problem 3. What if I have an n1 by n2 by... by nk grid. Now each switch toggles bulbs which agree in exactly k-1 co-ordinates.

Problem 4. We have the same grid as problem 3 but now the switches toggle bulbs agreeing with our bulb in exactly l coordinates for general l. 

I won't say how far we got with these at the meetup because it would spoil the fun somewhat. I will say that only problem 1 actually occurs in Beardon's excellent book, so it's not to be taken as a given that there are nice solutions beyond that. 

The condom problem

Today I'm going to share a problem which I learned from Prof Vishnu Jejjela . As usual I won't spoil the solution for a few days so that people can mull over it. In the mean time I'll lay out the problem and a few special cases which I think are worth thinking about.

n men and m women want to have sex with each other. In particular they want all mn pairings to occur. The problem is that they all have a different STD and they all want to avoid getting anyone else's STD. For this they have condoms, these condoms have some fairly well defined properties.

1. They can be nested.
2. They can be used in either direction.
3. They never break.
4. Diseases never get through them.
5. On the other hand if a side of condom A is in contact with a side of condom B with disease x then disease x jumps to condom A.
6. If a person with disease y uses condom C then disease y appears on the side of condom C actually used.

The problem of course is to minimize the number of condoms used. The naive solution of course involves using mn condoms (1 for each encounter).

Two classically special interesting cases are
1. m=n=2.
2. n=1, m an arbitrary odd integer.

Happy New Year!!!

Happy new year guys (yes both of you). Anyway it's 2015 and I'm going to try to be a more active blogger this year (yes, yes easy to say at the start of the year when everything else going on is comparatively quite).

Anyway what better way to kick of the year than with the 2015 problem of the year . I've played with it a bit but I think I'll leave the ones I have out for a bit. There are a few nice tricks which I don't really want to spoil for anyone reading. I'll leave it for a commenter to spoil. :-)

Problem of the year 5775

OK so it's 5775 according to the Hebrew calendar. Which means of course that it's Problem of the year time (well ok it's a bit late but so what).

5775 turns out to be kinda tough (or I'm being silly).

(5+7)/(7+5)=1 is pretty easy.
getting 5-sqrt(7+7-5)=2 took me quite a while.

I have at the time of writing no way to get 3.

5-(7/7)^5=4
5*((7/7)^5)=5
5+(7/7)^5=6

all sort of fit together. 

7 like 3 I don't have yet, although I've spent less time on it.

5+sqrt(7+7-5)=8

5-(7/7)+5=9
5*(7/7)+5=10
 5+(7/7)+5=11

 again fit into a nice group.

This is about where I stopped thinking about these (because 3 and 7 where incomplete).

 So naturally the point of this post is to see what people come up with for the 3,7 and things bigger than 11.  Enjoy!

SATMO 2014

Last Saturday, the 3rd annual South African Tertiary Mathematics Olympiad was held. For US based readers think of an SA version of the Putnam contest.  Full results, questions and solutions,  can be found here.

Two of the questions (8 and 17) this year where submitted by me. Question 8 I first heard from Michael Lugo and question 17 I first heard from Pravesh Ranchod. 

A huge thank you for organizing the event goes out to Stephan Wagner of Stellenbosch Univesity, as does a huge congrats to the winners.

I've recently been involved in the formation of the Wits Undergrad Maths Society

This has involved going to a bunch of lectures and handing out sign up sheets. It's also involved a lot of talking to lecturers and other organizers to timetable this but I want to focus on the signup sheets.

Turns out that a fair number of students use there wits e-mail addresses and a fair number use something else (g-mail the most common). By inspection there seems to be a very very large correlation between what sheet you signed and which e-mail you put down. that is to say that some of the sheets are almost completely (or in some smaller cases completely) covered with students using there wits e-mail while others are almost devoid of student e-mails. 

Presumably students look at the first few people putting down a wits e-mail address and decide to copy or see the first few haven't and decide to use the non-wits address. Anyone have any thoughts on what causes this?

Hats

Dmytro Yeroshkin and myself have uploaded a paper to the archive here .
It's on a  hats game which is well described here by Tanya Khovanova, and originally proposed by Lionel Levine

There are a lot of opportunities for future work left. In particular the original problem with alot of wise men instead of the 2 player case described in our paper.