Difference between revisions of "KGS math club"
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|Consider the two player game that begins with an even length sequence of positive integers. Each player, in turn, removes either the first or last of the remaining integers, ending when all the integers have been removed. A player's score is the sum of the integers that they removed; the winner is the player with the higher score (with a tie if equal scores). Show that Player One has a non-losing strategy, i.e., can always force a tie or a win. | |Consider the two player game that begins with an even length sequence of positive integers. Each player, in turn, removes either the first or last of the remaining integers, ending when all the integers have been removed. A player's score is the sum of the integers that they removed; the winner is the player with the higher score (with a tie if equal scores). Show that Player One has a non-losing strategy, i.e., can always force a tie or a win. | ||
|style="background-color:rgb(220,230,255);" | [[KGS math club/hints_1_1|hints]] [[KGS math club/solution_1_1|solution]] [[KGS math club/solution_1_2|solution2]] | |style="background-color:rgb(220,230,255);" | [[KGS math club/hints_1_1|hints]] [[KGS math club/solution_1_1|solution]] [[KGS math club/solution_1_2|solution2]] | ||
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| + | |30.6.2008 | ||
| + | |style="background-color:rgb(220,230,255);" | amkach | ||
| + | |For n >= 2, consider the n-dimensional hypercube with side length 4 centered at the origin of n-space. Place inside of it 2^n n-dimensional hyperspheres of radius 1, centered at each of the points (+-1, +-1, ..., +-1). These hyperspheres are tangent to the hypercube and to each other. | ||
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| + | Then place an n-dimensional hypersphere, centered at the origin, of size so that it is tangent to each of the 2^n hyperspheres of radius 1. In which dimensions n is this central hypersphere contained within the hypercube? | ||
| + | |style="background-color:rgb(220,230,255);" | [[KGS math club/solution_3_1| solution ]] | ||
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<!-- TEMPLATE, COPY-PASTE-FILL-IN: date, author, problem, solver; n = problem number, m = solution number, then click the red link and copypaste the solution, save. There. If anyone can do the wiki table formatting more elegantly, be my guest; after all, this is wikiiii. | <!-- TEMPLATE, COPY-PASTE-FILL-IN: date, author, problem, solver; n = problem number, m = solution number, then click the red link and copypaste the solution, save. There. If anyone can do the wiki table formatting more elegantly, be my guest; after all, this is wikiiii. | ||
Revision as of 14:37, 30 June 2008
A group of people on Kiseido Go Server Mathematics room.
The meaning of this page is to collect the problems posed there and save hints and solution suggestions.
| Date | Author | Problem | Solutions |
|---|---|---|---|
| 20.2.2007 | StoneTiger | Does any member of the sequence 1, 4, 20, 80, ... generated by x(n) = 6x(n-1) - 12x(n-2) + 8x(n-3) ever have a factor in common with 2007? | sigmundur |
| 21.6.2008 | amkach | Consider the two player game that begins with an even length sequence of positive integers. Each player, in turn, removes either the first or last of the remaining integers, ending when all the integers have been removed. A player's score is the sum of the integers that they removed; the winner is the player with the higher score (with a tie if equal scores). Show that Player One has a non-losing strategy, i.e., can always force a tie or a win. | hints solution solution2 |
| 30.6.2008 | amkach | For n >= 2, consider the n-dimensional hypercube with side length 4 centered at the origin of n-space. Place inside of it 2^n n-dimensional hyperspheres of radius 1, centered at each of the points (+-1, +-1, ..., +-1). These hyperspheres are tangent to the hypercube and to each other.
Then place an n-dimensional hypersphere, centered at the origin, of size so that it is tangent to each of the 2^n hyperspheres of radius 1. In which dimensions n is this central hypersphere contained within the hypercube? |
solution
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