On 02/02/2020 18:56, Keith F. Lynch wrote:
There have been no posts in response to either of my mystery tables for over a week. Is anyone still working on them, or shall I reveal what they are tables of? Here they are again:
[SNIP: first table, now solved]
1 2 3 4 5 6 7 8 1: 1.000000 1.545362 2.293678 3.240906 4.248364 5.259193 6.267074 7.272747 2: 1.545362 2.000000 2.576905 3.343752 4.270644 5.262645 6.267506 7.272793 3: 2.293678 2.576905 3.000000 3.589622 4.367036 5.288115 6.272582 7.273628 4: 3.240906 3.343752 3.589622 4.000000 4.596536 5.380654 6.299598 7.279833 5: 4.248364 4.270644 4.367036 4.596536 5.000000 5.600888 6.389624 7.307724 6: 5.259193 5.262645 5.288115 5.380654 5.600888 6.000000 6.603882 7.395990 7: 6.267074 6.267506 6.272582 6.299598 6.389624 6.603882 7.000000 7.606068 8: 7.272747 7.272793 7.273628 7.279833 7.307724 7.395990 7.606068 8.000000
The same thing happened with this one as with the other: I was writing up my failure to solve it when I solved it :-). At first glance it _looks_ like some sort of generalized mean thing: f(x,y) = F^-1((F(x)+F(y))/2) for some increasing function F, preferably increasing quite fast so that e.g. f(1,8) ~= 8. And it looks as if the very simple choice F(x) = exp(x) produces numbers quite similar to those in the table; small modifications, such as F(x) = sinh(0.9x) or better still sinh(x) / x^(2/3), do even better; but my admittedly crude attempts at playing with the shape of F have been unable to get the mean squared error below about 0.015, which suggests that maybe the answer _isn't_ of this form at all. (Only weakly; I haven't tried _very hard_ to optimize F.) But it turns out that the entries in the table satisfy the following (defining) equation: [... spoiler space ...] [... spoiler space ...] [... spoiler space ...] [... spoiler space ...] [... spoiler space ...] [... spoiler space ...] [... spoiler space ...] [... spoiler space ...] [... spoiler space ...] if the entry in position (x,y) is z, then x^z + y^z = 2z^z. It is not obvious to me whether this is actually equivalent to some sort of generalized mean as defined above. I'd guess not. What of course _is_ true is that z = ((x^z+y^z)/2)^1/z, so each entry is _a_ "power mean" of its row and column positions ... but with an exponent that equals the entry itself :-). -- g