Triangles with digits
“Never steal, but if you steal (a T-shirt), steal from the best.” – Woody Allen
(many thanks to Lee Sallows for this)
Today we would like to produce a sequence S where any triplet of successive digits – seen as side lengths – could form a triangle. Giorgos Kalogeropoulos helped:
S = 1, 2, 21, 22, 3, 4, 5, 6, 7, 8, 9, 28, 72, 65, 24, 32, 23, 31, 33, 13, 34, 25, 42, 43, 35, 36, 44, 14, 41, 441, 442, 45, 26, 52, 54, 46, 37, 53, 55, 15, 51, 551, 552, 56, 27, 62, 66, 16, 61, 661, 662, 67, 38, 63, 57, 39, 73, 64, 47, 48, 58, 49, 68, 59, 69, 74, 75, 76, 77, 17, 71, 771, 772, 78, 29, 82, 87, 79, 83, 86, 84, 85, 88, 18, 81, 881, 882, 89, 91, 99, 19, 92, 98, 93, 97, 94, 96, 95, 515, 524, ...
If you look indeed the first yellow triplet hereunder, you will see that the digits 1, 2 and 2 can form a triangle:
S = 1, 2, 21, 22, 3, 4, 5, 6, 7, 8, 9, 28, 72, 65, 24, 32, 23, 31,...
This is also true for this yellow triplet:
S = 1, 2, 21, 22, 3, 4, 5, 6, 7, 8, 9, 28, 72, 65, 24, 32, 23, 31,...
And for this one:
S = 1, 2, 21, 22, 3, 4, 5, 6, 7, 8, 9, 28, 72, 65, 24, 32, 23, 31,...
As usual, we want S to be the lexicographically earliest sequence of distinct terms having this property.
In forming S, we must always check that the digit d, squeezed between the digits a, b and y, z (see below) is a solution of the three inequalities [a+b>d], [b+y>d] and [y+z>d]:
a b d y z
Computing by myself the first terms of S was a nightmare. Gladly came Giorgos – Ευχαριστώ πολύ Γιώργο πολύ καλή δουλειά!
GK
> Here are the first 560 terms:
Commentaires
Enregistrer un commentaire