Map Coloring

Another fun challenge from Decision Management Community. A canonical example of the constraint programming (over finite domains) paradigm leads to a simple and elegant SQL solution.

The task is to color a map of six European countries: Belgium, Denmark, France, Germany, Luxembourg, and the Netherlands in such a way that no neighboring countries use the same color. You need to use no more than 4 colors: blue, red, green, or yellow.

Starting with two sets: country and color; who borders whom, and the rule.

country {BE, DK, FR, DE, LU, NL}
color   {B, R, G, Y}

BE borders {FR, LU, DE, NL}
DK borders {DE}
FR borders {BE, LU, DE}
DE borders {FR, LU, BE, NL, DK}
LU borders {FR, DE, BE}
NL borders {BE, DE}

Neighboring countries can not use the same color.

Reasoning

There are six countries, each one can have one of four colors, so in total there is a maximum of 4^6 = 4096 combinations to examine. The strategy is simple:

  1. Create a set of all possible combinations.
  2. Apply constraints.
  3. Present all results that satisfy given constraints.

The SQL code is straightforward and easy to follow. Try it, there are 144 possible solutions.

Variation

In a variation of the problem rules are changed a bit. Only three colors are allowed and some neighboring countries are allowed to have the same color, but there is a penalty cost assigned. The task is to find solutions with minimum penalty cost.

country {BE, DK, FR, DE, LU, NL}
color   {R, G, B}

These neighboring countries can use the same color,
with assigned cost:

LU-FR  $257
LU-DE  $904
LU-BE  $568

Reasoning v2

This time there are 3^6 = 729 combinations to examine. The strategy:

  1. Create a set of all possible combinations.
  2. Apply mandatory constraints.
  3. Calculate the cost for all tuples satisfying mandatory constraints.
  4. Find the minimum cost.
  5. Present all results that have the minimum cost.

Again, SQL code is easy to follow, this time there are 12 possible solutions.

Monkey Business

A challenge by Decision Management Community.

A zoo has four monkeys, during the monkey lunchtime each one ate a different fruit in their favourite resting place. Sam, who doesn’t like bananas, likes sitting on the grass. The monkey who sat on the rock ate the apple. The monkey who ate the pear didn’t sit on the tree branch. Anna sat by the stream but she didn’t eat the pear. Harriet didn’t sit on the tree branch. Mike doesn’t like oranges.

  1. What kind of fruit each monkey ate?
  2. Where their favourite resting place was?

So, how to solve this in SQL? It is easy to identify domains, predicates and rules; after that it is straightforward. Continue reading “Monkey Business”

Murder Mystery

Another fun puzzle, based on the problem No. 55 in [PFJ86] and published as a challenge by Decision Management Community.

Someone in Dreadsbury Mansion killed aunt Agatha. Agatha, the butler, and Charles live in Dreadsbury Mansion, and are the only ones to live there. A killer always hates, and is no richer than his victim. Charles hates no one that Agatha hates. Agatha hates everybody except the butler. The butler hates everyone not richer than aunt Agatha. The butler hates everyone whom Agatha hates. No one hates everyone. Who killed Agatha?

The idea — as in previous posts — is to use concept of predicates, constraints, relations, and Continue reading “Murder Mystery”

Reindeer Ordering

The Decision Management Community posted a fun challenge: Santa’s elves are supposed to order nine reindeer according to a set of rules. The rules are:

  1. Comet behind Rudolph, Prancer, and Cupid.
  2. Blitzen behind Cupid.
  3. Blitzen in front of Donder, Vixen, and Dancer.
  4. Cupid in front of Comet, Blitzen, and Vixen.
  5. Donder behind Vixen, Dasher, and Prancer.
  6. Rudolph behind Prancer.
  7. Rudolph in front of Donder, Dancer, and Dasher.
  8. Vixen in front of Dancer and Comet.
  9. Dancer behind Donder, Rudolph, and Blitzen.
  10. Prancer in front of Cupid, Donder, and Blitzen.
  11. Dasher behind Prancer.
  12. Dasher in front of Vixen, Dancer, and Blitzen.
  13. Donder behind Comet and Cupid.
  14. Cupid in front of Rudolph and Dancer.
  15. Vixen behind Rudolph, Prancer, and Dasher.

The challenge is to create a decision model, but Continue reading “Reindeer Ordering”