# Chapter 3: The Lost Logicians (puzzle hints and solutions)

#### Puzzle 1 – Treetops

1. All trees must be connected to form one network.
2. All bridges join trees horizontally or vertically and no two bridges may cross.
3. The numbers signify how many bridges link to that tree.
4. You may have up to two bridges running between any two trees.

“Solve the puzzle and find the safe way to traverse these treetops.”

#### Hint 1

“The puzzle is meant to challenge those seeking to reach the temple. Start by considering trees with a 1 on them, bearing in mind all trees must form one network.”

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#### Hint 2

“If you find yourself with too little information, notice that a tree with lots of connections will most likely have at least one bridge on each side. For example a tree with 7 would have to have at least one bridge on each side.”

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#### Puzzle 2 – Eyes

We have an ancient tradition upheld for centuries. If, at any time, you realise for certain that you have blue eyes, you must leave the camp that same night. There is a catch however. You may not in any way convey information about anyone’s eye colour to anyone else and you will struggle to find any reflective surfaces on this entire island.

“I must say I find your tradition most interesting, particularly since I can see at least one of you with blue eyes.” Isabelle stares lasers at Terry as he says this.

“You fly in thin air toucan, now my nerves are far from settled.” She huffs angrily and turns to look at you with a little less venom. “Say, maybe you can tell me if any of us will have to leave this place? Will anyone realise they have blue eyes? If so, how long do I have until people leave?”

You look around and see dozens of anxious faces watching you. Among them, you notice exactly 10 pairs of blue eyes. Can you work out if any of them will ever know for certain they have blue eyes after what Terry said? If so, when will they leave? As you bring out a pen and paper, Terry chimes in, “Keep in mind that every tribe member is perfectly logical.”

#### Hint 1

“Start with a simplified version of the problem. What would happen if only one of the tribe members had blue eyes?”

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#### Hint 2

“Once you’ve solved what happens when one person has blue eyes. Can you think about what would happen if two people had blue eyes? It may help to think about what each of those people can see and think back to what would have happened if there was only one person with blue eyes.”

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#### Hint 3

“Now consider three people with blue eyes. Is there a similarity between this and the case with 2 people? Remember that as soon as someone knows for certain they have blue eyes, they would leave that night and the other islanders would notice their absence the next day.”

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#### Hint 4

“Just as important as people leaving is the fact that people with blue eyes have not yet left. Can you gather any information from this?”

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#### Puzzle 1 (Treetops) – Solution

We start by using both of Terry’s hints, specifically to look at trees with a 1 and those with a 3  with only two connecting sides. Then continue drawing bridges, keeping in mind which trees already have all of their connections.

This is an example of a Hashi puzzle which again you can look up online to find more of.

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#### Puzzle 2 (Eyes) – Solution

The Tribe problem ultimately relies on inductive reasoning. We will use a blue dot to symbolize a blue eyed person, and a brown dot otherwise (eye colours other than blue may all be considered the same).

If there were only one blue eyed member, after Terry says “I can see at least one blue eyed person”, the blue person would look around and see nothing but brown eyes and deduce that they must have blue eyes themselves. They will then leave on the first night.

This statement in bold is crucial. Now let us consider the case with two blue eyed people below.

Since everyone can already see at least 1 other blue eyed person, no one can know for certain that they have blue eyes the moment Terry speaks. Thus, everyone wakes up on day 2 and no one has left. Now, suppose you are person A.

Person A only sees one other blue eyed person and thinks: If I had brown eyes, person B would have left last night. Since everyone is still here, it must mean that person B can also see someone with blue eyes and hence, person A deduces they have blue eyes. Likewise, person B deduces they also have blue eyes. Both blue eyed members leave on the second night.

Now this reasoning will work for any number of blue eyed people. Suppose we have 10 blue eyed tribe members. Each tribe member can see at least 9 blue eyed people. If after 9 nights, everyone is still present, it means that there must be 10 blue eyed people. Thus, if you only see 9, you deduce that you yourself are the 10th person with blue eyes. Thus, every tribe member with blue eyes will leave on the 10th night.

The reasoning in this problem is very subtle and the only knowledge the islanders can gather comes in fact from the lack of knowledge the other islanders have up to that point. If this problem interests you, you may like to think about the following extension problems:

1. What did Terry say, that each islander didn’t already know?
2. What happens if one of the islanders is blind? (Here, blind means their eyes are open as normal but they don’t know the eye colours of anyone else. They will of course be told verbally if people have left the camp).
3. What happens if more than one islander is blind? There may be various different outcomes depending on how many blue eyed people are blind.