Alphaα

Betaβ Eta

ηZetaζ

Gammaγ

Deltaδ Epsilon

ε

Thetaθ

Iotaι

Kappaκ

The Battle for

Abbey Ridge

Alphaα

Betaβ Eta

ηZetaζ

Gammaγ

Deltaδ Epsilon

ε

Thetaθ

Iotaι

Kappaκ

The Abbey mountain range is the key battle ground between two mighty armies.

The range is made up of 10 ridges named after the first 10 letters in the greek alphabet.

Which ever army controls the most ridges will win the battle, and the war.

Alphaα

Betaβ Eta

ηZetaζ

Gammaγ

Deltaδ Epsilon

ε

Thetaθ

Iotaι

Kappaκ

Each army will advance up the ridge during the cover of darkness and battle will commence at daybreak.

The side with the most forces at the top of each ridge will win the battle. There is no luck involved.

If there are the same number of legions at the top of a ridge, it is a stalemate, and counts nothing for either side.

Alphaα

Betaβ Eta

ηZetaζ

Gammaγ

Deltaδ Epsilon

ε

Thetaθ

Iotaι

Kappaκ

This is a fair fight. Each general has exactly 100 legions to deploy.

Alphaα

Betaβ Eta

ηZetaζ

Gammaγ

Deltaδ Epsilon

ε

Thetaθ

Iotaι

Kappaκ

Only skill, strong leadership and anticipating your opponents strategy will lead to victory.

Alphaα

Betaβ Eta

ηZetaζ

Gammaγ

Deltaδ Epsilon

ε

Thetaθ

Iotaι

Kappaκ

45

50

1

11

11

25

2525

25

Look at these two generals deployment. What do you think ?

Why is blue’s deployment hopeless ?

Is red much better ?

Alphaα

Betaβ Eta

ηZetaζ

Gammaγ

Deltaδ Epsilon

ε

Thetaθ

Iotaι

Kappaκ

45

50

1

11

11

25

2525

25

You have been fully trained and briefed. Now it is time to fight.

Alphaα

Betaβ Eta

ηZetaζ

Gammaγ

Deltaδ Epsilon

ε

Thetaθ

Iotaι

Kappaκ

Let battle commence.

45

50

1

11

11

25

2525

25

αβ

ηζγ

δε

θι κ

Tactics And Strategies

αβ

ηζγ

δε

θι κ

• The brain struggles initially with the distribution of 100 units if it overcomplicates things.

• Hence many new players go for a weak 10 x 10 strategy.

10α

10β

10η

10ζ

10γ

10δ

10ε

10θ

10ι

10κ

• Why is this weak ?

• 10 is the average value, but you will probably need at least 6 above average (strong) armies to win.

Difficulty: Easy

αβ

ηζγ

δε

θι κ

• Consider this distribution.

11α

11β

11η

11ζ

11γ

11δ

11ε

11θ

11ι

1κ

• We now have nine stronger than average armies, by creating one weak army.

• The 9 x 11 + 1 army should beat the 10 x 10 army 9 -1

9 x 11 = 99 99 + 1 = 100

Difficulty: Easy

αβ

ηζγ

δε

θι κ

• The next time we fight an improving opponent they will probably have realised they must create slightly stronger armies. Hence we must create better, if fewer, armies.

12α

12β

12η

12ζ

12γ

12δ

12ε

12θ

2ι

2κ

• We now have eight stronger than average armies by creating two weak armies.

•This distribution should beat the 10 x 10 army 8 – 2.

•This distribution should beat the 9 x 11 + 1 army 9 – 1.

8 x 12 = 96 96 + 2 + 2 = 100

Difficulty: Medium

• As an opponent improves we must improve this strategy of stronger armies. Here is a flow chart of gradually stronger armies.

10 10 101010 10 10 10 10 10

• Once we reach 6 x 14 + 4 x 4 the game becomes even more tactical.

11 11 111111 11 11 11 11 1

12 12 121212 12 12 12 2 2

13 13 131313 13 13 3 3 3

14 14 41414 14 14 4 4 4

Better

Difficulty: Medium

• Many good players like to win outright with 6 big armies.

• This means the smaller armies are “useless” and so they are sacrificed.

• The final 4 x17 +2 x 16 army is better than the 6 x16 + 4 x 1 army as it normally draws, but wins if gets its 16’s against the opponents 1’s.

14 14 41414 14 14 4 4 4

Better

15 15 31515 15 15 3 2 2

16 16 11616 16 16 1 1 1

17 17 01617 17 16 0 0 0

16 16 11616 16 16 1 1 1

17 17 16017 17 0 16 0 0

Difficulty: Medium

• In order to think fast we need to know the following arithmetic.

10 x 10 = 100

9 x 11 = 99

8 x 12 = 96

6 x 13 = 78 7 x 13 = 91

6 x 14 = 84 7 x 14 = 98

6 x 15 = 90

5 x 16 = 80 6 x 16 = 96

5 x 17 = 85

5 x 18 = 90

5 x 19 = 95

5 x 20 = 100

Difficulty: Hard

•Let us assume we are fighting a skilled opponent who plays.

• To beat this distribution we need to play stronger armies than 17.

17 17 01617 17 16 0 0 0

Difficulty: Hard

18 18 2218 18 18 2 2 2

• We now know we are guaranteed a draw with 5 strong army wins, but the weak armies will probably also generate some victories. The 5 x 18 + 5 x 2 army should win 9 – 1

•Let us assume we are fighting a very skilled opponent who plays.

• To beat this distribution we need to play stronger armies than 18, while keeping some weak armies stronger than 2.

Difficulty: Very Hard

20 20 5020 20 0 5 5 5

• As we now have 8 ridges with the potential to beat the

5 x 18 + 5 x 2 army we should win most of the time.

18 18 2218 18 18 2 2 2

•We are now playing a very strong opponent who plays

• We now realise he has over reinforced his first four armies. So we should plan a defence which has most chance of picking off all his weak armies.

Something like,

Difficulty: Brain Ache

10 10 101010 10 10 10 10 10

• That’s right – The “weakest” deployment wins.

20 20 5020 20 0 5 5 5

• To sum things up in a diagram we have

Difficulty: Brain Ache

10 101010

10 10 1010 10 10

16 16416

16 16 164 4 4

20 2050

20 20 05 5 5

Better

Better

Better

• We have analysed the game into a circular argument.

• i.e if A beats B, and B beats C, that doesn’t guarantee A will beat C. ( A bit like football!)

•Mathematicians call this property “Nontransative” and you can read more about it here.

Nontransitive dice - Wikipedia, the free encyclopedia

Alphaα

Betaβ Eta

ηZetaζ

Gammaγ

Deltaδ Epsilon

ε

Thetaθ

Iotaι

Kappaκ

Plenary

Let every member of the class pick there best deployment and let the computer make “everyone play everyone” to see who has the best.

Win = 2 points. Draw = 1 point. Loss = 0 points.

Rossett_Ridge_Class_DK.xls