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National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine Team Selection Test
2016 Ukraine Team Selection Test
1
1
Part of
2016 Ukraine Team Selection Test
Problems
(1)
(6n+3)-game
Source: 2016 Ukraine TST
7/20/2018
Consider a regular polygon
A
1
A
2
…
A
6
n
+
3
A_1A_2\ldots A_{6n+3}
A
1
A
2
…
A
6
n
+
3
. The vertices
A
2
n
+
1
,
A
4
n
+
2
,
A
6
n
+
3
A_{2n+1}, A_{4n+2}, A_{6n+3}
A
2
n
+
1
,
A
4
n
+
2
,
A
6
n
+
3
are called holes. Initially there are three pebbles in some vertices of the polygon, which are also vertices of equilateral triangle. Players
A
A
A
and
B
B
B
take moves in turn. In each move, starting from
A
A
A
, the player chooses pebble and puts it to the next vertex clockwise (for example,
A
2
→
A
3
A_2\rightarrow A_3
A
2
→
A
3
,
A
6
n
+
3
→
A
1
A_{6n+3}\rightarrow A_1
A
6
n
+
3
→
A
1
). Player
A
A
A
wins if at least two pebbles lie in holes after someone's move. Does player
A
A
A
always have winning strategy?Proposed by Bohdan Rublov
combinatorics
game
polygon