Veterinaria y Zootecnia VZ012 Investigación de
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A Game for Two
Here is a single-board game for two. You can play on the city plan shown here or draw a larger version for yourself.
You need two coins, one for the cop , the o ther for the robber. Start with each coin on its picture. The rules are simple : The cop always moves first. After that , the players take turns to move. You move a coin one block only , left or right, up or down- that is, from one comer to the next. The aim is for the cop to catch the robber, which is d one by the cop land ing on the robber on his move. To m ake the game interesting , the cop must catch the robber in 20 moves, or he loses.
HINT : There is a way for the cop to nab the robber. The secre t lies in the top left comer of the city plan .
Sprouts
A Game for Two
S p routs is one of the best of the really new p aper-and-pencil games. A C am bridge ( England) m athem atician invented it in the 1 9 60s. Its name comes from the shapes y ou end up with . I t is a game of topology. a branch of m athem atics which is very briefly explained in "The Bridges of Konigs berg" ( page 2 5 ) . Topology is the geom etry of flop py rulers . wiggly lines, an d stretchy sheets of paper!
This is how Sprouts is played. On a clean
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sheet of p aper begin by drawing three or four sp ots. We'll work with three spots.
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E ach of the two play ers takes turns at j oin- ing the sp ots with lines, which can be as wiggly as y ou like. You must put a new spot somewhere along that line.
No lines m ay cross.
Y ou can d raw a line from a spot back onto
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itself to m ake a loop - with , of course, a new sp ot on it .A sp ot is "dead " when it has three lines lead ing to it ; no m ore lines can connect to it. To sh ow it is dead , put a stroke through it or shade it in.
The winner is the one who draws the last line . A good way to win is to trap "live" sp ots inside loops so that y our opponent can n ot use them .
M athem aticians h ave worked out how m any m oves the game can go on for: The number lies between twice an d 3 times the number of sp ots you start with. Starting with 3 sp ots , the game can go on for be t ween 6 and 9 m ove s ; with 4 starting spots,
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between 8 and 1 2 m oves. And so on. But nobody h as pro ved th is yet ! Here is a sample game. In it player A wins in 7 m oves.
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POINTS
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B p lays
Mo rra
A Game for Two
This very old finger game comes from Italy . One player is called Morra. On a given signal-a nod of the head , for example- both p layers put up either o ne or two fingers both at the same time. The ru les are in summary form :
B oth play ers show same number of fingers : M orra wins two pennies Morra two fingers, opp onent one finger: M orra loses one penny M orra one finger, opponent two fingers : M orra loses three pennies S ee if y ou can fin d a strategy that cuts Morra 's losses or even lets him win. The best strategy is given in the answer section.
Hex
A Game for Two
Only recently invented in Denmark , this is a m arvelous gam e , which is also called Black and White. It seems absurdly simple but is open to very cun n ing play or strategy . as it is called . The game is played on a diamond shap ed b o ard m ade up of either hexagons, hence the name, or triangles.
The board usually has I I hex agons ( or triangles) on each side. Two oppo site sides of the d iamond are Black 's side ; the o ther two are White's . The hex agons at the comer of the board belong to either player. The players take turns m arking hex agons. White m arks his p oint with a circle , Black with a heavy blob . The aim is to connect opposite sides of the board with
an unbroken line of dots or blobs. (On the triangle paper, t wo dots are adj acent if there is a link ing line between them.)
The first player to m ake an unbroken line is the winner. Two lines c annot cross, so there can never be a draw. Mathematicians h ave proved that the first player can always win , but they don't say how he is to do so! You can buy special p rinted paper with a grid of hexagons or triangles printed on it. If y ou draw up your own board , as here , do so in ink and play in pencil lightly so you can rub out the circles after each gam e.
To learn some of the strategies of Hex , play a game on a 2-by-2 board with just four hexagons. The player who makes the first move obviously wins. On a 3-by-3 board the first player wins by m aking his first move in the center of the board . This is because the first player has a double play on both sides of his opening cell , so his opponent h as no way to keep him from winning in the next two moves. On a 4-by-4 board ( see pictu re)
things are more complicated. The first player will win if he play s in one of the four num bered cells, but if he plays in any other cell , he can always be defeated. For an I I -by-I l b oard , as shown , the play is far too complicated to be analyzed .