## Monday, 25 August 2014

### Dangerous caterpillars solution

 Figure 1. The usual suspects.
Last time I asked if we could work out which caterpillars from Figure 1 are safe based on the following rules:
1. If a caterpillar has blue eyes or spots it is dangerous. If a caterpillar has both, we can't tell if it is dangerous.
2. All safe caterpillars have more than one of the following features: teeth, blue eyes, spots, or spikes.
3. Caterpillars with both teeth and spots are dangerous.
As I suggested Venn diagrams are expertly suited to this kind of puzzle.

To solve the puzzle using Venn diagrams we, first, decide what our sets are going to be. In the clues there are 4 features that are discussed: teeth, blue eyes, spots and spikes. Thus, we construct a Venn diagram of these four qualities and insert the caterpillars to their appropriate spaces. This can be seen in Figure 2. Note that in this specific problem we have to use the full Venn diagram form of the four sets, rather than the Euler diagram that we discussed, previously.
 Figure 2. Separating the caterpillars into types.
Now that we have our diagram we use the clues to remove sections and restrict the safe caterpillar possibilities.

The simplified Venn diagrams corresponding to each of the clues can be seen in Figure 3. For example, the first clue states that animals with blue eyes and spots (but, perhaps, not both) are dangerous. Thus, we remove the sections corresponding to the blue eyed caterpillars and the spots. However, the region over which they intersect is left, because we can't be sure if those caterpillars are dangerous or not.
 Figure 3. Removing sections of the Venn diagram based on the clues. Left: the blanked out section corresponds to either having blue eyes or spots, but not both. Center: removing the sections with only one feature. Right: removing the section with teeth and spots.
This process is carried out for all three clues. Finally, we put together only the sections that remain in all three diagrams in Figure 3 to get Figure 4.
 Figure 4. Only one caterpillar left.
As you see will from Figure 4 there are only 3 sections of the Venn diagram left. These include two sections in the blue eyes and spots region. Because of rule 1, we cannot tell if those sections are safe, or not. Luckily, there are no caterpillars in these regions, so no risks need be taken. The only caterpillar left has both spikes and teeth... would you trust this guy?