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The Vanishing Dots
Game for two players.
Put 20 dots on a piece of paper. The two players play alternately. Each player has to cross out
1. 2 or 3 dots in his playing turn. The player can vary the number of dots he crosses out in each
turn.
The winner is the player who crosses out the last dot.

• • • • • • • • • • • • • • • • • • • •
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Joe crosses out like this/. Gary crosses out like this\ .

Joe Gary Joe Gary Joe Gary Joe Gary Joe Gary

Gary said that by choosing not to play first he should always win.
How can that be? Is it true? Find out by playing.

The Take It Away Game
Game for two players.
1 5 matches or counters are placed between the two players.

<::) C> c:>
c:>o oc:,o
Oc,c,c,
OC>C>

Then each. in turn. takes 1 2 or 3 counters.

C)
Gary o c::::> 3 0000 2 C) C> Joe
C> 0 C, C)
C> C)

Gary C) c:::, 0 2 c:::, c::, C) 1 Oc, Joe
c::,
C)C> c:>oo
C)

c:, 2 2
Gary C) 0 0 C> C> Oc::,e> Joe
C) C>o
0 0 C)

C> C> 0 2 1 0
Gary 0 C) C> none c::., <::> C> Joe
oo C> left C> C>

The winner is the one who ends up with an odd number of counters in front of him. Gary won
the game. Joe thinks it is a swindle and that the first player can always win . Is that true?
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