(3, 3, 12) | = | (1, 8, 9) |

(4, 8, 9, 21) | = | (3, 7, 14, 18) |

(4, 7, 21, 36) | = | (1, 12, 27, 28) |

(5, 7, 10, 14, 27) | = | (3, 6, 15, 18, 21) |

(5, 85, 85, 169, 425) | = | (13, 17, 125, 289, 325) |

(17, 21, 24, 48, 54, 238) | = | (3, 4, 14, 119, 126, 136) |

**A**
You may have noticed that for each pair the products contain
the same prime factors, though they aren't equal.
What *is* equal is the product of x^{x}:

3^{3} 3^{3} 12^{12} |
= | 1^{1} 8^{8} 9^{9} |

4^{4} 8^{8} 9^{9} 21^{21} |
= | 3^{3} 7^{7} 14^{14} 18^{18} |

4^{4} 7^{7} 21^{21} 36^{36} |
= | 1^{1} 12^{12} 27^{27} 28^{28} |

5^{5} 7^{6} 10^{10} 14^{14} 27^{27} |
= | 3^{3} 6^{6} 15^{15} 18^{18} 21^{21} |

5^{5} 85^{85} 85^{85} 169^{169} 425^{425} |
= | 13^{13} 17^{17} 125^{125} 289^{289} 325^{325} |

17^{17} 21^{21} 24^{24} 48^{48} 54^{54} 238^{238} |
= | 3^{3} 4^{4} 14^{14} 119^{119} 126^{126} 136^{136} |

This was suggested by one of the exhibits
at the Ramanujan museum at SASTRA University in Kumbakonam, India,
which I saw at the Dec.2003 conference where the museum was inaugurated.
Ramanujan found several examples, including (3,3,12)=(1,8,9).
I used LLL (the algorithm of Lenstra, Lenstra, and Lovasz
for lattice basis reduction) to find the rest.
The example (5,85,85,169,425) = (13,17,125,289,325)
illustrates that all numbers involved can be odd
(and indeed coprime to 6). The examples
(5,27,28,70) = (6,15,49,60), (1,6,8,20,30) = (2,2,12,24,25),
and (with all numbers odd) (3,21,21,25,63) = (5,7,27,45,49)
have equal sums, products of x,
*and* products of x^{x}.

This use of LLL is the same trick I had used to find “subset anagrams”
— see Alex Healy's note “Finding Anagrams Via Lattice Reduction”
here,
and Alex's interactive implementation.
My talk at the AMS/MAA joint meeting in 2003
about this and other novel applications of lattice reduction
was written up in *Science* by Barry Cipra
(Vol. **299** (2003), 650-651).