Integers becoming right
triangles
Hello Math-Fun,
Write an integer somewhere on a grid,
one digit per square:
+---+---+---+---+---+---+
|
| | |
| | |
+---+---+---+---+---+---+
|
| | |
| | |
+---+---+---+---+---+---+
|
| 1 | 3 | 4 | 0 | |
+---+---+---+---+---+---+
|
| | |
| | |
+---+---+---+---+---+---+
|
| | |
| | |
+---+---+---+---+---+---+
|
| | |
| | |
+---+---+---+---+---+---+
The digits will now move vertically:
- odd digit j will climb j squares
- even digit k will sink k squares
- zero stays where it is
In our example, the digits of 1340 move
like this (their former position is in a whiter
shade of pale):
+---+---+---+---+
|
| 3 | |
|
+---+---+---+---+
|
| | |
|
+---+---+---+---+
| 1 | |
| |
+---+---+---+---+
| 1 | 3 | 4 | 0 |
+---+---+---+---+
|
| | |
|
+---+---+---+---+
|
| | |
|
+---+---+---+---+
|
| | |
|
+---+---+---+---+
|
| | 4 | |
+---+---+---+---+
Encircle now those new digits with an
elastic band:
The resulting shape is not a right triangle – which is a pity
because we are precisely looking for integers turning into right triangles! The
additional constraint being that ALL the new digits must be ON the elastic
band.
Here are a few good examples:
Integer 130 (the right angle is on 1):
Integer 313333 (right angle = 1):
Integer 2201022 (right angle = 1):
Here is a bad example:
Integer 21000 (yes we have a triangle,
yes there is a right angle --on 1--, but the band
doesn’t pass through ALL digits):
Questions:
a) Find the 40 first integers producing
a right triangle
b) Find the biggest integer producing a
right triangle
c) Find the smallest integer producing a
rectangle.
[Answers here in December 2010]
Best,
É.
__________
(a):
S = 101, 103, 120, 130, 210, 301, 1032,
1307, 2301, 3210, 6120, 7031, 10325, 11311, 13079, 19753, 22022, 22422, 24680, 33133,
33533, 35791, 43210, 44244, 44644, 52301, 55355, 55755, 64208, 66466, 66866,
75319, 77577, 77977, 80246, 86120, 88688, 91357, 97031, 99799, ...
(b) as found by
Nicolas Graner and Sébastien Perez-D., the last term of
the above finite sequence is a 291-digit integer:
“9999...99989 with 289 times a 9 in front, right angle on the 8.” [Seb]
Or, as Nicolas puts it in French:
Je
propose : « Neuf cent quatre-vingt-dix-neuf octoquadragintillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf septenquadragintillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf sesquadragintillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf quinquaquadragintillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf quattuorquadragintillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf tresquadragintillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf duoquadragintillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf unquadragintillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf quadragintillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf noventrigintillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf octotrigintillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf septentrigintillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf sestrigintillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf quinquatrigintillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf quattuortrigintillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf trestrigintillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf duotrigintillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf untrigintillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf trigintillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf novemvigintillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf octovigintillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf septemvigintillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf sesvigintillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf quinquavigintillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf quattuorvigintillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf tresvigintillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf duovigintillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf unvigintillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf vigintillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf novendécillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf octodécillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf septendécillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf sedécillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf quinquadécillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf quattuordécillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf tredécillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf duodécillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf undécillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf décillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf nonillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf octillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf septillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf sextillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf quintillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf quatrillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf trillions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf billions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-dix-neuf millions
neuf cent quatre-vingt-dix-neuf mille neuf cent quatre-vingt-neuf ».
Soit 289 fois le chiffre 9, puis 89. C’est ça ? Pour le (c) je propose 103210, ou
012301 si on accepte un 0 initial.
Perfect, Nicolas... And as we don’t accept
leading zeroes, yes, the answer (c) is:
(c) 103210
(smallest integer producing a rectangle).
Thanks to
all contributors,
Best,
É.