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):

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.

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.