Archive for partition

Le Monde puzzle [#1650]

Posted in Books, Kids, R with tags , , , , , , , , , on September 5, 2018 by xi'an

A penultimate Le Monde mathematical puzzle  before the new competition starts [again!]

For a game opposing 40 players over 12 questions, anyone answering correctly a question gets as reward the number of people who failed to answer. Alice is the single winner: what is her minimal score? In another round, Bob is the only lowest grade: what is his maximum score?

For each player, the score S is the sum δ¹s¹+…+δ⁸s⁸, where the first term is an indicator for a correct answer and the second term is the sum over all other players of their complementary indicator, which can be replaced with the sum over all players since δ¹(1-δ¹)=0. Leading to the vector of scores

worz <- function(ansz){
  scor=apply(1-ansz,2,sum)
  return(apply(t(ansz)*scor,2,sum))}

Now, running by brute-force a massive number of simulations confirmed my intuition that the minimal winning score is 39, the number of players minus one [achieved by Alice giving a single good answer and the others none at all], while the maximum loosing score appeared to be 34, for which I had much less of an intuition!  I would have rather guessed something in the vicinity of 80 (being half of the answers replied correctly by half of the players)… Indeed, while in SIngapore, I however ran in the wee hours a quick simulated annealing code from this solution and moved to 77.

And the 2018 version of Le Monde maths puzzle competition starts today!, for a total of eight double questions, starting with an optimisation problem where the adjacent X table is filled with zeros and ones, trying to optimise (max and min) the number of positive entries [out of 45] for which an even number of neighbours is equal to one. On the represented configuration, green stands for one (16 ones) and P for the positive entries (31 of them). This should be amenable to a R resolution [R solution], by, once again!, simulated annealing. Deadline for the reply on the competition website is next Tuesday, midnight [UTC+1]

Le Monde puzzle [#1033]

Posted in Books, Kids, R with tags , , , , on December 19, 2017 by xi'an

lemondapariA simple Le Monde mathematical puzzle after two geometric ones I did not consider:

  1. Bob gets a 2×3 card with three integer entries on the first row and two integer entries on the second row such that (i) entry (1,1) is 1, (ii) summing up subsets of adjacent entries produces all integers from 1 to 21. (Adjacent means sharing an index.) Deduce Bob’s voucher.
  2.  Alice gets Bob’s voucher completed into a 2×4 card with further integer entries. What is the largest value of N such that all integers from 1 to N are available through summing up all subsets of entries?

The first question only requires a few attempts but it can be solves by brute force simulation. Here is a R code that leads to the solution:

alsumz<-function(sol){return( 
  c(sol,sum(sol[1:2]),sum(sol[2:3]),sum(sol[4:5]),
  sum(sol[c(1,4)]), sum(sol[c(1,5)]),sum(sol[1:3]),
  sum(sol[c(1,4,5)]),sum(sol[c(1,2,5)]),
  sum(sol[c(2,4,5)]), sum(sol[c(1,2,3,5)]),sum(sol[2:5]), 
  sum(sol[c(1,2,4)]),sum(sol[c(1,2,4,5)]),sum(sol[1:4]),
  sum(sol)),sum(sol[c(2,3,5)]))}

produces (1,8,7,3,2) as the only case for which

(length(unique(alsum(sol)))==21)

The second puzzle means considering all sums and checking there exists a solution for all subsets. There is no constraint in this second question, hence on principle this could produce N=2⁸-1=255, but I have been unable to exceed 175 through brute force simulation. (Which entitled me to use the as.logical(intToBits(i) R command!)

CRiSM workshop on estimating constants [slides]

Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , , , , on May 4, 2016 by xi'an

A short announcement that the slides of almost all talks at the CRiSM workshop on estimating constants last April 20-22 are now available. Enjoy (and dicuss)!

CRiSM workshop on estimating constants [#2]

Posted in pictures, Statistics, Travel, University life, Wines with tags , , , , , , , , , on March 31, 2016 by xi'an

The schedule for the CRiSM workshop on estimating constants that Nial Friel, Helen Ogden and myself host next April 20-22 at the University of Warwick is now set as follows. (The plain registration fees are £40 and accommodation on the campus is available through the online form.)

April 20, 2016
11:45 — 12:30: Adam Johansen
12:30 — 14:00: Lunch
14:00 — 14:45: Anne-Marie Lyne
14:45 — 15:30: Pierre Jacob
15:30 — 16:00: Break
16:00 — 16:45: Roberto Trotta
17:00 — 18:00: ‘Elevator’ talks
18:00 — 20:00: Poster session, Cheese and wine

April 21, 2016
9:00 — 9:45: Michael Betancourt
9:45 — 10:30: Nicolas Chopin
10:30 — 11:00: Coffee break
11:00 — 11:45: Merrilee Hurn
11:45 — 12:30: Jean-Michel Marin
12:30 — 14:00: Lunch
14:00 — 14:45: Sumit Mukherjee
14:45 — 15:30: Yves Atchadé
15:30 — 16:00: Break
16:00 — 16:45: Michael Gutmann
16:45 — 17:30: Panayiota Touloupou
19:00 — 22:00: Dinner

April 22, 2016
9:00 — 9:45: Chris Sherlock
9:45 — 10:30: Christophe Andrieu
10:30 — 11:00: Coffee break
11:00 — 11:45: Antonietta Mira

CRiSM workshop on estimating constants [#1]

Posted in pictures, Statistics, Travel, University life with tags , , , , , , , , , on January 19, 2016 by xi'an

The registration for the CRiSM workshop on estimating constants that Nial Friel, Helen Ogden and myself host next April 20-22 at the University of Warwick is now open. The plain registration fees are £40 and accommodation on the campus is available through the same form.

Since besides the invited talks, the workshop will host two poster session with speed (2-5mn) oral presentations, we encourage all interested researchers to submit a poster via the appropriate form. Once again, this should be an exciting two-day workshop, given the on-going activity in this area.