August 2017 - Challenge

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Ponder This Challenge:

This month's challenge is dedicated to the memory of Prof. Maryam Mirzakhani, who passed away on July 15, 2017. Prof. Mirzakhani was the first (and as of today, the only) woman to win a Fields Medal. Her husband, Jan Vondrak, worked at IBM Research from 2009-2015.

The challenge is to place at least eight black balls and one white ball on integer coordinates of a 4-by-6 billiard table (from (0,0) at the top left of the table to (4,6) at the bottom right) with four pockets at its corners, such that one can play all the black balls into the pockets one at a time, by hitting them with the white ball. As in billiards, the balls must bank from the rails with mirror symmetry. When the white ball hits a black ball, it replaces that black ball's location, and the black ball continues the natural motion. In this process, black balls must never hit another ball.
Please supply your answer as five lines of seven characters and a list of directions.
A direction is a speed vector represented as set of two numbers. The first is the horizontal component of the speed (positive to the right) and the second is the vertical component (positive is downwards).
For example, here is a solution with three balls:

(1,-1) (-1,-1) (-2,-1)

The first hit (direction 1,-1) collides with a black ball, which then banks three times and falls into the top left pocket.
The second hit (direction -1,-1) banks three times and hits a black ball into the top right pocket.
The third hit (direction -2,-1) puts the last black ball into the top left pocket.

We assume that the balls are of zero size.

Update (2/8): The idea was not to place balls on the rails, and even then you can place more than 8; allowing placing the balls on the rails let you place many more. A bonus '*' will be awarded for the best solutions for each case.

We will post the names of those who submit a correct, original solution! If you don't want your name posted then please include such a statement in your submission!

We invite visitors to our website to submit an elegant solution. Send your submission to the

If you have any problems you think we might enjoy, please send them in. All replies should be sent to:

Challenge: 31/07/2017 @ 12:00 PM EST
Solution: 02/09/2017 @ 12:00 PM EST
List Updated: 04/09/2017 @ 12:00 PM EST

People who answered correctly:

**Oleg Vlasii (31/07/17 15:28 IDT)
Steven Langerwerf & Susan Brommer (31/07/17 03:48 PM IDT)
Robert Gerbicz (31/07/17 03:45 PM IDT)
Serge Thill (31/07/17 07:25 PM IDT)
Joseph DeVincentis (31/07/17 08:32 PM IDT)
Kang Jin Cho (31/07/17 09:41 PM IDT)
Dan Dima (31/07/17 10:35 PM IDT)
George Barnett (01/08/17 01:08 AM IDT)
Siddhant Kothari (01/08/17 03:34 AM IDT)
*Yan-Wu He (01/08/17 04:20 AM IDT)
Jesse Rearick (01/08/17 05:40 AM IDT)
*Amos Guler (01/08/17 10:14 AM IDT)
Aviv Nisgav (01/08/17 11:16 PM IDT)
Shirish Chinchalkar (02/08/17 12:53 AM IDT)
Emir Haleva (02/08/17 10:54 AM IDT)
Alper Halbutogullari (02/08/17 12:59 AM IDT)
Erik Lagerstedt (02/08/17 13:49 PM IDT)
Chuck Carroll (02/08/17 02:39 PM IDT)
Zachary Foster (02/08/17 02:43 PM IDT)
Serino Dario (02/08/17 04:48 PM IDT)
Rogerio Ponce da Silva (02/08/17 05:47 PM IDT)
Shaleen Deep (02/08/17 06:21 PM IDT)
Emir Haleva (02/08/17 10:54 AM IDT)
**Tyler Mullen (02/08/17 10:23 PM IDT)
Vladimir Volevich (03/08/17 10:24 AM IDT)
Harald Bögeholz (03/08/17 10:56 AM IDT)
*Gary M. Gerken (03/08/17 01:45 PM IDT)
Muralidhar Seshadri (04/08/17 12:23 AM IDT)
Loukas Sidiropoulos (04/08/17 12:15 PM IDT)
Graham Hemsley (04/08/17 01:10 PM IDT)
Boris Silantyev (04/08/17 03:19 PM IDT)
Tamir Ganor & Shouky Dan (04/08/17 03:26 PM IDT)
Felix Huguenin (04/08/17 10:19 PM IDT)
Aashish Diayar (05/08/17 01:30 PM IDT)
Alexander Shay Geller (05/08/17 06:21 PM IDT)
Michael Bruhn (06/08/17 03:11 AM IDT)
Franciraldo Cavalcante (06/08/17 04:54 AM IDT)
JJ Rabeyrin (06/08/17 04:59 PM IDT)
Paul Lucero (06/08/17 05:00 PM IDT)
Slavisa Pajkanovic (06/08/17 08:39 PM IDT)
Yuval Snappir (07/08/17 08:00 AM IDT)
Francis Golding (06/08/17 10:48 PM IDT)
*Paolo Farinelli (08/08/17 05:44 AM IDT)
**Uoti Urpala (08/08/17 06:09 AM IDT)
*Andrei Pilka (09/08/17 01:32 PM IDT)
**Dmitry Kim (12/08/17 06:04 PM IDT)
**Florian Fischer (13/08/17 01:30 PM IDT)
Eden Saig & Yuval Nezri (13/08/17 04:10 PM IDT)
*Todd Will (16/08/17 11:00 PM IDT)
*Andreas Knüpfer (16/08/17 11:31 PM IDT)
Mathieu Parent (18/08/17 08:18 PM IDT)
*David Greer (21/08/17 11:49 PM IDT)
Colton Graham (22/08/17 04:31 PM IDT)
Ranjit Eswaran (23/08/17 05:57 AM IDT)
Hakan Summakoğlu (23/08/17 05:20 PM IDT)
*Nyles Heise (26/08/17 07:42 AM IDT)
Karthik Kakarla (28/08/17 06:04 AM IDT)
Hakan Summakoğlu (27/08/17 10:47 PM IDT)
Ed Hall (28/08/17 02:10 AM IDT)
Oscar Volpatti (29/08/17 08:03 AM IDT)
*Leonid Kalmankin (31/08/17 10:22 AM IDT)
John Van Ryneveld (31/08/17 01:52 PM IDT)
*Motty Porat (01/09/17 12:17 AM IDT)