# Chapter 1: Stranded (puzzle hints and solutions)

#### Puzzle 1 – Canals

“Can you find a way of drawing a canal in the sand so that the waves can reach the fish spawn? You are to dig a single continuous canal without forming any loops, forks or leaving canals isolated.”

#### Hint 1

“It may help to cross off anywhere you know there can’t be a canal”

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#### Hint 2

“Use the fact that the canals must all be connected and can’t have forks to continue what the frog started.”

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#### Hint 3

“Focus on the numbers 3 and 0. For the two 3s next to each other, can you work out the only way the canal must go around them? Where must 2 of the 3 canals go for the 3 in the corner?”

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#### Puzzle 2 – Figs

“My puzzle for you is this. You may conduct rounds of testing in which you give each of the four frogs a collection of chunks from any of the figs you choose. All four of them will then eat all of the chunks in one go and those who ate rotten fruit will turn bright red before fading back to green. Can you find the least number of rounds needed to identify which is the rotten fig? How will you assign the chunks in each round?”

#### Hint 1

“It may be faster to feed more than one chunk to some frogs in the first round, despite not being able to tell which of the chunks were rotten if that frog goes red.”

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#### Hint 2

“How many possible outcomes are there to any single round of testing? Remember that each frog can remain green or turn red. Does this hint at a possible theoretical minimum number of rounds required?”

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#### Hint 3

“It seems to me as though there are exactly 16 possible outcomes, since each frog can be one of two states and there are four of them. Can you find a link between this problem and binary numbers so that each outcome points to a specific fig?”

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#### Puzzle 1 (canals) – Solution

You should have found that the only way to save the fish spawn and keep the frog happy is with the following arrangement. We include some explanations for each step. Find more of this type of puzzle by searching Slitherlink into a search engine.

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#### Puzzle 2 (figs) – solution

You may have found a way of finding the rotten fig in two rounds as follows:

Divide the figs into four piles of four and feed each pile to a single frog. One of the frogs will then turn red, narrowing down the possible rotten figs to just four. We can then feed one of these figs to each frog and whoever turns red must have eaten the rotten fig!

However as Terry hinted, there are 16 possible outcomes to any round, with each of the four frogs having two outcomes. So we could try to find a bijection, a 1-1 mapping, between possible outcomes and each fig.

One way of doing this is using binary numbers. Using 4 bits, 1s or 0s, we can represent any number from 0 up to 15 in base 2. A 1 in a column means we must add the corresponding power of 2 to our total to find what the number is in base 10. See below for an example: