Zoombini Maths - Basic Sorting in the Stone Cold Caves

Tackling some simple Carroll diagrams in the Stone Cold Caves (Not So Easy and Oh So Hard modes)

The second puzzle on the Zoombinis' journey is the Stone Cold Caves, and another based mostly on sorting and sets. There are four caves, guarded by two pairs of rock trolls. The two smaller trolls on the furthest left and right determine whether a particular Zoombini can travel up that path, while the two larger central trolls determine whether a Zoombini can enter their two caves. These entry requirements are based on the characteristics of the Zoombinis, in a similar way to the Allergic Cliffs.

The Mathematical Learning notes that accompany the game list identical skills to the Allergic Cliffs:

• observation
• forming and testing theories
• forming sets
• using evidence
• logical reasoning

As with the Cliffs, I tried to spot the strategies I was using to solve each difficulty level. One strategy that was quite efficient was choosing Zoombinis that looked fairly similar to follow each other - this allowed me to work out which characteristics were important in each attempt at the puzzle.

​Not So Easy
Initially, the puzzle is quite straightforward to solve; one of the vertical trolls will not allow any Zoombinis in at all, while one of the side trolls only allows Zoombinis with a particular characteristic - in the example below, the bottom troll won't let anyone in, and the right-hand troll will only admit Zoombinis with blue noses. This ends up being mathematically identical to the Allergic Cliffs on Not So Easy mode, albeit with another two redundant options.

After annotating the white lines over the puzzle, I realised that it had started to look suspiciously like a Carroll diagram, a two-way table used for grouping objects which have or do not have a particular attribute or set of attributes. Here, we have "blue noses" and "not blue noses" as our vertical characteristics and "Zoombinis" and "not-Zoombinis" (or in other words, nothing) as our horizontal characteristics. This similarity to Carroll diagrams becomes much more explicit on the next two difficulty levels, but falls apart slightly for Very Very Hard.

Oh So Hard
The next difficulty level unlocks all four caves, although it doesn't necessarily follow that each cave will have Zoombinis in each time - this depends on the party make-up. This time, one of both the horizontal and vertical pair of trolls will only allow Zoombinis with a particular characteristic, while the Zoombinis without this characteristic get sent up the other path.

When all four caves are used, it's easier to see the similarities with a traditional 2x2 Carroll diagram, in this case splitting into smooth/not smooth hair down the columns and propeller/not propeller along the rows.

Although this puzzle sorts Zoombinis in a similar way to the Allergic Cliffs on Very Hard (namely by picking two characteristics from two types that are important), the layout of the puzzle as a grid of four means there is direct correlation with each section of the Venn diagram. The top left quadrant represents the intersection - those Zoombinis with smooth hair and propellers. Unlike the Allergic Cliffs, the sole Zoombini sitting in the intersection here is not mixed in with the other groups. 

We can also distinctly see the sets of Zoombinis with only one of the key features - top right has propellers but not smooth hair, while bottom left has smooth hair but not propellers. Finally, the bottom right quadrant contains all the Zoombinis without either of the key features.

The next post will look at the complexities that arise in this puzzle on Very Hard and Very Very Hard mode. I think there's definitely scope for using this puzzle along with the Allergic Cliffs to examine Venn diagrams in more detail, particularly as you can see the intersection and sets "A and not B" and "B and not A" so clearly.

P.S. In other news, I found my original game manual today! There's nothing revolutionary in there, but I had a little bit of a nostalgic moment having a flick through, and realising just how far computers have come since 1996 - original system requirements included a 33MHz processor, 2MB disc space and Quick Time V2. It's astounding to think that a game that barely ran on the computers when I was at primary school now runs easily on a mobile phone...

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