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.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
Chess and Mathematics Conference London Olympia 6-7 Dec
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Re: Chess and Mathematics Conference London Olympia 6-7
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Re: Chess and Mathematics Conference London Olympia 6-7
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.).
(You'd hope they'd give you the marks if you explained how you read it and produced a logical answer.).
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Re: Chess and Mathematics Conference London Olympia 6-7
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.
Oops, arithmetical error. 296 it is.
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Re: Chess and Mathematics Conference London Olympia 6-7
If you regard matrix formulae as simple, this fits the bill for a board of m rows x n columns :-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.
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
Code: Select all
35 0 89 0 103 0 69 0
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
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Re: Chess and Mathematics Conference London Olympia 6-7 Dec
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.
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.
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Re: Chess and Mathematics Conference London Olympia 6-7 Dec
Freddie Illingworth is a junior international bridge player.
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.
Joe Benton was a member of Richmond Junior Chess Club about 10 years ago.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.
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.