l’affriolé, Paris

sunrise on the Cam

Le Monde puzzle [#755?]

Le Monde puzzle of last weekend was about sudoku-like matrices.

Consider an (n,n) matrix containing the integers from 1 to n². The matrix is “friendly” if the set of the sums of the rows is equal to the set of the sum of the columns. Find examples for n=4,5,6. Why is there no friendly matrix when n=9?

Checking for small n’s seems easy enough:

friend=function(n){
s=1
while (s>0){
  A=matrix(sample(1:n^2),ncol=n)
  s=sum(abs(sort(apply(A,1,sum))-
        sort(apply(A,2,sum))))}
A
}

For instance, running

> friend(3)
     [,1] [,2] [,3]
[1,]    8    4    2
[2,]    1    9    5
[3,]    6    3    7
> friend(4)
     [,1] [,2] [,3] [,4]
[1,]   14   10   11    6
[2,]   13    3   12    8
[3,]    4    5   16   15
[4,]    9    1    2    7
> friend(5)
     [,1] [,2] [,3] [,4] [,5]
[1,]   17   14   10   16    5
[2,]    8   19    7   15   22
[3,]    2    3   25   13    6
[4,]   11   12   18    1   21
[5,]   24   23   20    4    9
>friend(6)
     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]   14    4   36   30   10   18
[2,]    7   16   24   27   32   11
[3,]   21   25   12    5   22   17
[4,]   26   20    6   31   19   34
[5,]    3   35    1   28    9   29
[6,]   23    2   33   15   13    8

produces right answers. But the case n=6 proved itself almost too hard for brute-force handling!!! I have no time to devise a simulated-annealing code to speed up the resolution so this will have to wait till the next weekend edition of Le Monde. As to why n=9 does not enjoy a solution… (When n=2 there is no solution, as proven by the brute force exhibition of all cases.)

Cambridge skyline

GPUs in computational statistics

The workshop in Warwick yesterday went on very quickly! The room was packed. The three first talks were by myself, Christophe and Pierre, so less about GPUs and more about simulation techniques which could benefit from or even require implementation on GPUS. (I did manage to have complete slides this time… More seriously, Christophe’s talk set me thinking on the issue of estimating the likelihood function in ways different (?) from the one used in ABC.) The second half was more novel for me, in that the three talks went into the computing and computer aspects of GPUS, with Chris Holmes doing sparse [Lasso-like] regression on a scale otherwise [i.e. w/o GPUs] impossible, Chris [fourth Chris in the list of speakers!] Barnes explaining ABC for molecular biology and design (a point I plan to discuss on a later post), with even more details about the architecture and programming of GPUs, and Michael Stumpf delivering a grand finale, with essentially three talks into one: network analysis (incl. terrific movie bits incorporated within the beamer slides!), GPUs vs. CPUs and older alternatives, and random generators on GPU, commenting on a recent paper by Salmon et al. (SC, 2011) and showing that true gains in efficiency from using GPUs involved a heavy involvement into the hardware structure… A very exciting day followed by Stilton cheese degustation and haggis (if not poems) to celebrate Burns’ night!

Some hae meat and canna eat,
And some wad eat that want it;
But we hae meat, and we can eat,
And sae let the Lord be thankit.