1. Alpha α Beta β Eta η Zeta ζ Gamma γ Delta δ Epsilon ε Theta θ Iota ι Kappa κ The Battle for Abbey Ridge

2. Alpha α Beta β Eta η Zeta ζ Gamma γ Delta δ Epsilon ε Theta θ Iota ι Kappa κ 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.

3. Alpha α Beta β Eta η Zeta ζ Gamma γ Delta δ Epsilon ε Theta θ Iota ι Kappa κ 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.

4. Alpha α Beta β Eta η Zeta ζ Gamma γ Delta δ Epsilon ε Theta θ Iota ι Kappa κ This is a fair fight. Each general has exactly 100 legions to deploy.

5. Alpha α Beta β Eta η Zeta ζ Gamma γ Delta δ Epsilon ε Theta θ Iota ι Kappa κ Only skill, strong leadership and anticipating your opponents strategy will lead to victory.

6. Alpha α Beta β Eta η Zeta ζ Gamma γ Delta δ Epsilon ε Theta θ Iota ι Kappa κ 45 50 1 1 1 1 1 25 Look at these two generals deployment. What do you think ? Why is blue’s deployment hopeless ? Is red much better ?

7. Alpha α Beta β Eta η Zeta ζ Gamma γ Delta δ Epsilon ε Theta θ Iota ι Kappa κ 45 50 1 1 1 1 1 25 You have been fully trained and briefed. Now it is time to fight.

8. Alpha α Beta β Eta η Zeta ζ Gamma γ Delta δ Epsilon ε Theta θ Iota ι Kappa κ Let battle commence. 45 50 1 1 1 1 1 25

9. α β η ζ γ δ ε θ ι κ Tactics And Strategies

10. α β η ζ γ δ ε θ ι 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. 10 α 10 β 10 η 10 ζ 10 γ 10 δ 10 ε 10 θ 10 ι 10 κ 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

11. α β η ζ γ δ ε θ ι κ Consider this distribution. 11 α 11 β 11 η 11 ζ 11 γ 11 δ 11 ε 11 θ 11 ι 1κ1κ 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

12. α β η ζ γ δ ε θ ι 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. 12 α 12 β 12 η 12 ζ 12 γ 12 δ 12 ε 12 θ 2ι2ι 2κ2κ 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

13. As an opponent improves we must improve this strategy of stronger armies. Here is a flow chart of gradually stronger armies. 10 Once we reach 6 x 14 + 4 x 4 the game becomes even more tactical. 11 1 12 22 13 333 14 4 444 Better Difficulty: Medium

14. 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 4 444 Better 15 3 322 16 1 111 17 01617 16000 1 111 17 16017 01600 Difficulty: Medium

15. 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 = 847 x 14 = 98 6 x 15 = 90 5 x 16 = 806 x 16 = 96 5 x 17 = 85 5 x 18 = 90 5 x 19 = 95 5 x 20 = 100 Difficulty: Hard

16. Let us assume we are fighting a skilled opponent who plays. To beat this distribution we need to play stronger armies than 17. 17 01617 16000 Difficulty: Hard 18 22 222 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

17. 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 50 0555 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 22 222

18. 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 That’s right – The “weakest” deployment wins. 20 50 0555

19. To sum things up in a diagram we have Difficulty: Brain Ache 10 16 4 444 20 50 0 555 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

20. Alpha α Beta β Eta η Zeta ζ Gamma γ Delta δ Epsilon ε Theta θ Iota ι Kappa κ 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