Chess and Mathematics Conference London Olympia 6-7 Dec

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Arshad Ali
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Re: Chess and Mathematics Conference London Olympia 6-7

Post by Arshad Ali » Tue Nov 04, 2014 1:49 pm

Matthew Turner wrote:Arshad,
I have read the BMO question a couple of times but I still cannot see why you are multiplying by 2.
Matt
My fault -- my eye elided over "one in each row." Otherwise you would also have to include all the paths that start from the left (a2, a4, a6, a8) and end up on the right (h1, h3, h5, h7). A symmetry argument suffices to conclude they must be same as the number of paths from bottom to top, but don't want to double count the long diagonal when adding both sets.

MartinCarpenter
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Re: Chess and Mathematics Conference London Olympia 6-7

Post by MartinCarpenter » Tue Nov 04, 2014 2:20 pm

Although being pedantic, 'which meet at their corners' could certainly be interpreted in a few different ways :)
(You'd hope they'd give you the marks if you explained how you read it and produced a logical answer.).

Arshad Ali
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Re: Chess and Mathematics Conference London Olympia 6-7

Post by Arshad Ali » Tue Nov 04, 2014 5:54 pm

I get 326 from two different methods of calculation. I'm fairly certain this is correct (or they could both be wrong and just happen to coincide). I'll write down my modus operandi when I have a bit of spare time.

Oops, arithmetical error. 296 it is.

E Michael White
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Re: Chess and Mathematics Conference London Olympia 6-7

Post by E Michael White » Sun Nov 30, 2014 11:32 pm

Matthew Turner wrote:296 is indeed the right answer
I find this interesting because if the board was infinitely wide there would be 2^9 possibilities (7 moves and 4 initial starting squares).
296 = 2^9 - 6^3 I find it incredibly hard to believe that is a coincidence. However, I am assured by very eminent mathematicians that this is indeed the case and that you cannot develop a (simple) formula for an x by y board.
If you regard matrix formulae as simple, this fits the bill for a board of m rows x n columns :-

N=Bx(T^(m-1))x[1]

N is the number of routes. ^ is to the power and x matrix multiplication
B is an n element bit map vector of start positions for the bishop on row 1 eg b1,d1,f1,h1 has B=(0 1 0 1 0 1 0 1)
[1] is an n element column vector composed of 1’s.
T is an nxn matrix with 1’s immediately above and below the diagonal and 0’s elsewhere.

Eg T :-
0 1 0 0 0 0 0 0
1 0 1 0 0 0 0 0
0 1 0 1 0 0 0 0
0 0 1 0 1 0 0 0
0 0 0 1 0 1 0 0
0 0 0 0 1 0 1 0
0 0 0 0 0 1 0 1
0 0 0 0 0 0 1 0

For a standard 8x8 board T^7 is :-

Code: Select all

0	14	0	14	0	6	0	1
14	0	28	0	20	0	7	0
0	28	0	34	0	21	0	6
14	0	34	0	35	0	20	0
0	20	0	35	0	34	0	14
6	0	21	0	34	0	28	0
0	7	0	20	0	28	0	14
1	0	6	0	14	0	14	0
So Bx(T^7) is

Code: Select all

35	0	89	0	103	0	69	0
and N=296

For boards up to about 16 x 16 simple arithmetic, as used earlier in the thread, is easiest but beyond that Excel, cutting and pasting and the function MMULT is probably faster

Neill Cooper
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Re: Chess and Mathematics Conference London Olympia 6-7 Dec

Post by Neill Cooper » Thu Jun 04, 2015 11:32 am

Last year's UK IMO (international Maths Olympiad) team were selecetd on their problem solving ability. Out of all school pupils the team chosen was:

Name, School
Joe Benton, St Paul’s School
Gabriel Gendler, Queen Elizabeth’s School
Frank Han, Dulwich College
Freddie Illingworth, Magdalen College School
Warren Li, Fulford School
Harvey Yau, Ysgol Dyffryn Taf

Neel Nanda (first reserve), Latymer School

The first four of these can be found on the ECF grading database, as can the first reserve. However none has an active grade, they are all 'inactive'. In most cases they were reasonably active chess players at primary school, and being good problem solvers were good at chess, but have since focused their problem solving on Maths rather than Chess.

Richard James
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Re: Chess and Mathematics Conference London Olympia 6-7 Dec

Post by Richard James » Thu Jun 04, 2015 2:21 pm

Freddie Illingworth is a junior international bridge player.
English Bridge Union wrote:His hobbies include competing in maths Olympiads for England, the only thing he could find geekier than playing bridge for England.
Joe Benton was a member of Richmond Junior Chess Club about 10 years ago.

My impression is that the children who excel in primary school chess clubs these days tend to be the academic high achievers who will go on to develop other interests while chess increasingly takes a back seat. Which is all the more reason why Neill's work in promoting secondary schools chess is so important.