## Tuesday, 22 January 2013

It started with a Twitter post by Mario Livio, who is head of public outreach at the Space Telescope Science Institute (the folks who operate the Hubble Telescope) but most of us know him better as a writer of popular math and science. I especially liked his book on the so-called Golden Ratio.

Mario's tweet was, "For buffs:118=14+50+54=18+30+70=15+40+63=21+25+72 and product of each triple is 37,800."

I got to wondering what was the smallest number that would have more than one partition into three parts so that the product of the three was the same. Turns out after some pencil scribbles, that 13 appears to be the smallest: 13= 1+6+6 = 2+2+9 and both have a product of 36.
Fourteen had two pairs. One with a product of 40 and the other with a product of 72 (I leave the actual values to the interested reader to find.)

Twenty-one seemed to be the smallest with three different pairs. Twenty-two and twenty three were interesting because they had the same product: 5+8+9 = 22, 4+9+10 = 23, but both have a product of 360. Twenty-five also produced a pair with a product of 360.

By thirty-nine I found what I believe is the smallest value with a three different partitions with the same product: 6 + 8 + 25 = 5 + 10+ 24 = 4 + 15 + 20; all three with a product 10 1200.