*All hints are included below so please scroll down slowly to avoid revealing them all at once*
**Solutions are given at the very bottom of the page**
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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*Scroll down for hint 2*
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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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*Scroll down for hint 3*
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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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*Scroll down for hints to the fig puzzle*
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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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*Scroll down for hint 2*
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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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*Scroll down for hint 3*
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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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*Scroll down for solution to the canal puzzle*
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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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*Scroll down for solution to the fig puzzle*
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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:
You can read more about binary numbers here.
So what if we number the figs 0 to 15 and imagine the 4 frogs as our 4 bits where if a frog goes red, it’s like a bit taking the value 1. Now for each fig, work out what its number is in binary, 12 would become 1100 for example. What we want is that the test result: red, red, green, green would indicate that the 12th fig was rotten. So we just need to feed chunks of the 12th fig exactly where the binary number 1100 has a 1. In the diagram below, the value of the frogs is shown next to them and each binary number should be read from the bottom up. A 1 indicates feeding a chunk of that fig to that frog.
Now we’re done! Whatever the outcome, we can recover exactly what number the rotten fruit is. We include an example below:
This idea was crucial in the invention of Hamming’s error correcting code, where a binary string contains a message followed by some digits which, if the message gets corrupted, can not only tell you there has been an error, but even where! Read more about it here.
TOM ROCKS MATHS 