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Cell["\<\
In the field of Artificial Intelligence there is a famous jug-filling puzzle \
that has been long been used as benchmark for AI progress. The following \
replicates an online interaction with Chat GPT-3.5 in Aug 2023.\
\>", "Text",
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 "   To measure out exactly 3 litres of liquid using a 5L and a 2-liter jug, \
you can follow these steps:"
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2. Pour the contents of the 5-liter jug into the 2-liter jug. This will leave \
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4. Pour the remaining 3 liters from the 5-liter jug back into the 2-liter \
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6. Pour the contents of the 5-liter jug into the 2-liter jug until it is \
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liquid, and the 5-liter jug will have 5-2=3 liters remaining\
\>", "Text",
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8. Transfer the remaining 3 liters from the 5-liter jug to the empty 2-liter \
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\>", "Text",
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10. Pour 1 liter from the 5-liter jug into the 2-liter jug, which already \
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Cell["\<\
By following these steps, you can accurately measure out 3 liters of liquid \
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\>", "Text",
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Cell["\<\
Chat 3.5 had in fact solved the problem by its second step but it then sets \
off on a superfluous detour sprouting physics-defying nonsense. \
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Cell["\<\
This particular hallucination has become a go-to example of the LLM\
\[CloseCurlyQuote]s  inherent \[OpenCurlyDoubleQuote]stupidity\
\[CloseCurlyDoubleQuote]. But how stupid is it? Doesn\[CloseCurlyQuote]t the \
power of LLM\[CloseCurlyQuote]s reside in its potential for interaction, for \
chat. 

A little back and forth leads to understanding the LLM\[CloseCurlyQuote]s \
knowledge gaps and the following prompt teaching it the requisite physics.\
\>", "Text",
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reasoning. You know that if a jug contains x litres and you use it to fill \
another jug containing y litres, with y less than x, then you will have x-y \
litres left in the first jug. You also know, that if you are asked to measure \
out x litres and this amount already appears in any jug, you do not need to \
include any more steps and immediately finish your answer without including \
any more steps. You use this knowledge to succinctly answer jug-filling \
questions with the smallest possible number of steps.\n\n",
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To measure out 3 liters using a 5-liter jug and a 2-liter jug, you can follow \
these steps:

1. Fill the 5-liter jug completely.
2. Pour water from the 5-liter jug into the 2-liter jug until the 2-liter jug \
is full. Now you have 3 liters left in the 5-liter jug.

Here is the step-by-step process:

1. Fill the 5-liter jug: (5, 0)
2. Pour from the 5-liter jug into the 2-liter jug: (3, 2)

At this point, you have exactly 3 liters in the 5-liter jug.

Therefore, the smallest possible number of steps to measure out 3 liters is 2 \
steps.\
\>", "ChatOutput",
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Cell["\<\
Often simplified, but \
\[OpenCurlyDoubleQuote]unnatural\[CloseCurlyDoubleQuote] problems like the \
one above are used to \[OpenCurlyDoubleQuote]trick\[CloseCurlyDoubleQuote] \
the LLM by attempting to show that its understanding is superficial like that \
of a stochastic parrot. Let\[CloseCurlyQuote]s see how this prompting equips \
the LLM to handle a more \[OpenCurlyDoubleQuote]natural example\
\[CloseCurlyDoubleQuote] that originally popularized these types of riddles. \
\
\>", "Text",
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reasoning. You know that if a jug contains x litres and you use it to fill \
another jug containing y litres, with y less than x, then you will have x-y \
litres left in the first jug. You also know, that if you are asked to measure \
out x litres and this amount already appears in any jug, you do not need to \
include any more steps and immediately finish your answer without including \
any more steps. You use this knowledge to succinctly answer jug-filling \
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Cell["\<\
To measure out exactly 4 liters using a 5-liter jug and a 2-liter jug, follow \
these steps:

1. Fill the 5-liter jug completely.
2. Pour water from the 5-liter jug into the 2-liter jug until the 2-liter jug \
is full. This leaves you with 3 liters in the 5-liter jug.
3. Empty the 2-liter jug.
4. Pour the 3 liters from the 5-liter jug into the 2-liter jug. This leaves \
you with 1 liter in the 5-liter jug.
5. Fill the 5-liter jug completely again.
6. Pour water from the 5-liter jug into the 2-liter jug until the 2-liter jug \
is full. Since there is already 1 liter in the 2-liter jug, you can only add \
1 more liter to it. This leaves you with exactly 4 liters in the 5-liter jug.

So, the steps are:

1. (5, 0) - Fill the 5-liter jug.
2. (3, 2) - Pour water from the 5-liter jug into the 2-liter jug.
3. (3, 0) - Empty the 2-liter jug.
4. (1, 2) - Pour the remaining 3 liters from the 5-liter jug into the 2-liter \
jug.
5. (5, 2) - Fill the 5-liter jug again.
6. (4, 0) - Pour water from the 5-liter jug into the 2-liter jug until the \
2-liter jug is full.

You now have exactly 4 liters in the 5-liter jug.\
\>", "ChatOutput",
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Cell[TextData[{
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The numbers in the previous example were used in the Hollywood movie, ",
 StyleBox["Die Hard With a Vengeance",
  FontSlant->"Italic"],
 " as protagonists John McClane and Zeus Carver are faced with solving this \
riddle within 30s to avoid a plot-ending explosion. "
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Cell[TextData[{
 "But LLM\[CloseCurlyQuote]s are trained on a non-trivial proportion of \
humanity\[CloseCurlyQuote]s output so it\[CloseCurlyQuote]s plausible that \
deeply buried in its model sits pop-culture\[CloseCurlyQuote]s mashed \
descriptions of movie plot-lines. It\[CloseCurlyQuote]s conceivable then, \
that successfully solving this puzzle might merely ",
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Cell[TextData[{
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capacity. Can it consistently solve the ",
 StyleBox["same",
  FontSlant->"Italic"],
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with parameters that have never previously appeared? How consistent will its \
performance be across different LLM\[CloseCurlyQuote]s or the same LLM but \
with different versions?"
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Cell["\<\
We might hope that versions later than ChatGPT 3.5  would perform as well if \
not better on these types of problems - perhaps to the point of not even \
needing physics-teaching prompts. Let\[CloseCurlyQuote]s test this out \
firstly on the latest OpenAI (ChatGPT-4 Turbo) release and subsequently on \
some of its recent snapshot releases.\
\>", "Text",
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Cell["I have 5L and 2L jugs. How do I measure out 4L?", "SideChat",
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 "To measure out exactly 4 liters using a 5-liter jug and a 2-liter jug, you \
can follow these steps:\n\n1. ",
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exactly 4 liters of water in the 5L jug!"
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   "Here are the steps to measure out 4L using a 5L and a 2L jug:\n\n1. Fill \
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Given this uncertainty, if you were aiming to build an expert system for \
solving bucket-filling problems you would want to isolate the relevant \
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the number crunching. But we can also use an LLM to suggest how this is done. \
Here we use the built-in code assistant.\
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I have two jugs containing x litres and y litres respectively and want to \
measure out  z litres. Can you provide a mathematical relationship between x, \
y and z to determine if this possible?\
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