You can get a glimpse of the gorgeous very easily. After all, you can work out the following sum in your head: 1 + 2 + 3 + 4 + 5 = ?

The answer is… 1 + 2 + 3 + 4 + 5 = 15. So that sum is example of this pattern: n₁:n₂ = sum(n₁..n₂). A simple computer program will soon supply other sums of consecutive numbers following the same pattern. I think these patterns based on the pair n₁ and n₂ are beautiful, so I’d call them fair pairs:

15 = sum(1..5)
27 = sum(2..7)
429 = sum(4..29)
1353 = sum(13..53)
1863 = sum(18..63)
3388 = sum(33..88)
3591 = sum(35..91)
7119 = sum(7..119)
78403 = sum(78..403)
133533 = sum(133..533)
178623 = sum(178..623)
2282148 = sum(228..2148)
2732353 = sum(273..2353)
3882813 = sum(388..2813)
7103835 = sum(710..3835)
13335333 = sum(1333..5333)
17016076 = sum(1701..6076)
17786223 = sum(1778..6223)

I went looking for variants on that pattern. If the function rev(n) reverses the digits of n, here’s n₁:rev(n₂) = sum(n₁..n₂):

155975 = sum(155..579)
223407 = sum(223..704)
4957813 = sum(495..3187)

I like that pattern, but it doesn’t seem beautiful like n₁:n₂ = sum(n₁..n₂). Nor does rev(n₁):n₂ = sum(n₁..n₂):

1575 = sum(51..75)
96444 = sum(69..444)
304878 = sum(403..878)
392933 = sum(293..933)
3162588 = sum(613..2588)
3252603 = sum(523..2603)
3642738 = sum(463..2738)
3772853 = sum(773..2853)
6653691 = sum(566..3691)
8714178 = sum(178..4178)

But rev(n₁):rev(n₂) = sum(n₁..n₂) is beautiful again, in a twisted kind of way:

97944 = sum(79..449)
452489 = sum(254..984)
3914082 = sum(193..2804)
6097063 = sum(906..3607)
6552663 = sum(556..3662)

Now try swapping n₁ and n₂. Here’s n₂:n₁ = sum(n₁..n₂):

204 = sum(4..20)
216 = sum(6..21)
20328 = sum(28..203)
21252 = sum(52..212)
21762 = sum(62..217)
23287 = sum(87..232)
23490 = sum(90..234)
2006118 = sum(118..2006)
2077402 = sum(402..2077)
2132532 = sum(532..2132)
2177622 = sum(622..2177)

Do I find the pattern beautiful? Yes, but it’s not as beautiful as n₁:n₂ = sum(n₁..n₂). The beauty disappears in n₂:rev(n₁) = sum(n₁..n₂):

21074 = sum(47..210)
21465 = sum(56..214)
22797 = sum(79..227)
2013561 = sum(165..2013)
2046803 = sum(308..2046)
2099754 = sum(457..2099)
2145065 = sum(560..2145)

And rev(n₂):n₁ = sum(n₁..n₂):

638 = sum(8..36)
2952 = sum(52..92)
21252 = sum(52..212)
23287 = sum(87..232)
66341 = sum(41..366)
208477 = sum(477..802)
2522172 = sum(172..2252)
2852982 = sum(982..2582)
7493772 = sum(772..3947)
8714178 = sum(178..4178)

Finally, and fairly again, rev(n₂):rev(n₁) = sum(n₁..n₂):

638 = sum(8..36)
125541 = sum(145..521)
207972 = sum(279..702)
158046 = sum(640..851)
9434322 = sum(223..4349)

The beauty’s back. And it has almost become self-aware. In rev(n₂):rev(n₁) = sum(n₁..n₂), each side of the equation seems to be looking at the other half as those it’s looking into a mirror.

Previously Pre-Posted (Please Peruse)…