MASTERMIND

Henning Thomas(joint with Benjamin Doerr,Reto Spöhel and Carola Winzen)

Henning Thomas Mastermind ETH Zurich 2012

Mastermind

Board game invented by Mordechai Meirovitz in 1970

Henning Thomas Mastermind ETH Zurich 2012

Mastermind

The “Codemaker” generates a secret color combination of length 4 with 6 colors,

The “Codebreaker” queries such color combinations,

The answer by Codemaker is , depicted by black pegs , depicted by

white pegs

The goal of Codemaker is to identify m with as few queries as possible.secret

query answer

Henning Thomas Mastermind ETH Zurich 2012

Mastermind with n slots and k colors

The “Codemaker” generates a secret color combination of length n with k colors,

The “Codebreaker” queries such color combinations,

The answer by Codemaker is , depicted by black pegs , depicted by

white pegs

The goal of Codemaker is to identify m with as few queries as possible.secret

query answer

Henning Thomas Mastermind ETH Zurich 2012

Mastermind with n slots and k colors

The “Codemaker” generates a secret color combination of length n with k colors,

The “Codebreaker” queries such color combinations,

The answer by Codemaker is , depicted by black pegs , depicted by

white pegs

The goal of Codemaker is to identify m with as few queries as possible.secret

query answer

This talk:Black PegMastermind

Henning Thomas Mastermind ETH Zurich 2012

Mastermind with n slots and k colors

The “Codemaker” generates a secret color combination of length n with k colors,

The “Codebreaker” queries such color combinations,

The answer by Codemaker is , depicted by black pegs , depicted by

white pegs

The goal of Codemaker is to identify m with as few queries as possible.secret

query answer

This talk:Black PegMastermind

What is the minimum number t = t(k,n) of queries such that there exists a

deterministic strategy to identify every secret color combination?

Henning Thomas Mastermind ETH Zurich 2012

Some Known Results & Our Results

[Knuth ’76], In the original board game (4 slots, 6 colors) 5 queries are optimal.

Henning Thomas Mastermind ETH Zurich 2012

Some Known Results & Our Results

[Knuth ’76], In the original board game (4 slots, 6 colors) 5 queries are optimal.

[Erdős, Rényi, ’63], Analysis of non-adaptive strategies for 0-1-Mastermind

In this talk: [Chvátal, ’83], Asymptotically optimal strategy for

using random queries [Goodrich, ’09], Improvement of Chvátals results

by a factor of 2 using deterministic strategyOur Result: Improved bound for k=n by combining Chvátal

and Goodrich

Henning Thomas Mastermind ETH Zurich 2012

Lower Bound

Information theoretic argument:

...

start

query 1

query 2

1 leaf

n leaves

n2 leaves

query t nt leaves

0 n

Henning Thomas Mastermind ETH Zurich 2012

Upper Bound (Chvátal)

Idea: Ask Random Queries. Intuition:

The number of black pegs of a query is Bin(n, 1/k) distributed.

Hence, we ‚learn‘ roughly bits per query. We need to learn n log k bits. t satisfies

0 n

Henning Thomas Mastermind ETH Zurich 2012

Comparison Lower Bound vs Chvátal

The optimal number of queries t satisfies

Problem for k=n: Non-Adaptive: Learning does not improve during

the game. For k=n we expect 1 black peg per query. We learn a constant number of bits. This yields

good ifk=o(n)

Henning Thomas Mastermind ETH Zurich 2012

Upper Bound (Goodrich)

Idea: Answer “0” is good since we can eliminate one

color from every slot!

Henning Thomas Mastermind ETH Zurich 2012

Upper Bound (Goodrich)

Implementation: Divide and Conquer1. Ask monochromatic queries for every color.

Obtain Xi = # appearances of color i.

2. Ask

3. Calculate Li = # appearnace of color i in left halfRi = # appearnace of color i in right

half

11 ... 1 22 ... 211 ... 1 33 ... 3

11 ... 1 kk ... k

b2

b3

bk

11 ... 122 ... 2

kk ... k

Henning Thomas Mastermind ETH Zurich 2012

Upper Bound (Goodrich)

Implementation: Divide and Conquer1. Ask monochromatic queries for every color.

Obtain Xi = # appearances of color i.

2. Ask

3. Calculate Li = # appearnace of color i in left halfRi = # appearnace of color i in right half

4.Recurse in the left and right half (without step 1)

5.Runtime for k=n:

11 ... 1 22 ... 211 ... 1 33 ... 3

11 ... 1 kk ... k

b2

b3

bk

11 ... 122 ... 2

kk ... k

Henning Thomas Mastermind ETH Zurich 2012

Comparison Lower Bound vs Goodrich

For k=n Goodrich yields

Problem: When Goodrich runs for a while, the blocks

eventually become too small that we cannot learn as many bits as we would like to.

Henning Thomas Mastermind ETH Zurich 2012

Combining Chvátal and Goodrich

Goodrich is good at eliminating colors. Chvátal is good for k << n.

Idea: 2 phases.

(i)Goodrich(ii)Chvátal

Henning Thomas Mastermind ETH Zurich 2012

Henning Thomas Mastermind ETH Zurich 2012