Note, that the question in the riddle section:
What 3 numbers give the same result when multiplied and added together? (Answer 1,2,3)
has of course and infinite number of other answers, such as -1,0,1
as -1+0+1=0 and -101=0
which both models have missed. In fact for any k which is integer the numbers -k,0,k will work.
Mathematica can find all of these solutions with
Solve[a + b + c == a*b*c, {a, b, c}, Integers]
(*{{c -> ConditionalExpression[(a + b)/(-1 + a b), Or[
And[
Element[
Alternatives[a, b, (a + b)/(-1 + a b)], Integers], a >= 1, b >
a^(-1)],
And[
Element[
Alternatives[a, b, (a + b)/(-1 + a b)], Integers], a >= 1, b <
a^(-1)],
And[
Element[
Alternatives[a, b, (a + b)/(-1 + a b)], Integers], b < a^(-1),
a <= -1],
And[
Element[
Alternatives[a, b, (a + b)/(-1 + a b)], Integers], a <= -1, b >
a^(-1)]]]}, {a ->
ConditionalExpression[0, (b | b) \[Element] Integers],
c -> ConditionalExpression[-b, (b | b) \[Element] Integers]}}*)
combining GPT-4 with the Mathematica Kernel via LLMTools like so
tool = LLMTool[{"runCode", "Run Wolfram Language code"}, "code",
If[ChoiceDialog["Run this code?\n\n" <> #code],
ToString[ToExpression[#code], InputForm], "Code not allowed"] &]
LLMSynthesize["What 3 Integers give the same result when multiplied \
and added together? Find all solutions. Use the tool.", All,
LLMEvaluator -> <|"Tools" -> tool|>]
This will produce a longer output, which after some tool call, contains the right answer.
Cheers,
Marco