Logic Masters Deutschland e.V.

Title: Bumper Pool
Type: Sudoku
Link: Penpa+ link to Bumper Pool You may use Normal/Large/Medium/Small Blue, Green, or Red numbers in any combination.
Variants: Renban

1) Sudoku

• Place the digits 1 through 9 once each in every row, column, and box.

2) Bumper Pool Basics

• 2.1) There are 2 sets of balls--5 red and 5 white. The 10 balls contain a complete set of digits 1 - 9 with a repeating 5. The digits on each set will add to the same sum.
• 2.2) Every ball contains one or more arrows. Take a shot for each arrow and only in the direction of the arrow. Although not needed for the solve, some arrows have multiple solutions. Not all arrows are given.
• 2.3) The digit to be placed in the ball's initial cell is determined by counting how many cells the ball touches on its way to, and including, its final cell; which will contain the same digit. For instance the 5 ball will touch the center of 5 cells; coming to rest on the center of the 5th cell--which would now be placed the digit 5. A ball may travel to a cell more than once. Any cell that contains green is table and a ball may occupy it.

3) Bumper Pool First Shot

• 3.1) Assume all rebounds and bounces occur at the center of each cell even though the boundary of the pool table is drawn on the edge of a cell. *I realize this is aesthetically unpleasant and stretches the realm of physics but it was a necessary compromise to keep the rules simple and make the bounces more intuitive.*
• 3.2) For each arrow, the initial shot can be a center shot or use "English".
• 3.3) A center aim shot reflects as follows:
• 3.3.1) If a ball enters a cell orthogonally and encounters an obstacle (wall or bumper), it bounces directly back from the direction it came.
• 3.3.2) If a ball enters a cell diagonally and encounters an obstacle, it bounces out at a 90 degree angle that is perpendicular to the incoming path.
• 3.3.3) If a ball encounters the 45 degree walls in boxes 1, 3, 7, and 9, it bounces away at 90 degrees if the ball entered the cell orthogonally, or bounces directly back if the ball entered the cell diagonally.
• 3.4) If a shot has English:
• 3.4.1) With left English the ball bounces back to the left at 45 degrees after encountering an obstacle.
• 3.4.2) With right English the ball bounces back to the right at 45 degrees after encountering an obstacle.
• 3.4.3) Balls cannot travel along the 45 degree wall in boxes 1, 3, 7, and 9. These are special cases. English shots will be the 45 degree angle between the normal 90 reflection and the initial inbound path after encountering an obstacle.
• 3.4.4) Only the most elite pool player can successfully shoot at any wall straight on and use English so this initial shot is not permitted for this puzzle to keep the shots somewhat simple. (Easter egg. I made this shot in the puzzle! Can you find it?)

4) Bumper Pool Additional Bounces

• 4.1) AFTER the initial shot, the ball ALWAYS rebounds in the same manner as a centered shot when it encounters an obstacle.
• 4.2) The ball may touch a specific obstacle (edge of the table or bumper) ONE TIME ONLY. That is to say, it can touch many bumpers but only touch the same bumper once.
• 4.3) The bumpers are small enough that a ball may occupy the same cell as a bumper.
• 4.4) If a ball falls into one of the two holes (black circles between the top and bottom bumpers), it is sunk and can no longer travel. For instance, if the 8 ball falls into a hole on the sixth cell of travel, that path is invalid as an eight.
• 4.5) A ball can travel through other balls. (The assumption is that all of the balls are moving as the game is being played)

5) Renban

• There is a 4 cell long pool cue made of the finest Renban wood. The four digits must be consecutive but in any order.

If Ball A has a value of 7, a valid cell could be R7C2. Invalid cells are R4C8 due to a 7 in the row and R7C6 because of a 7 in the column.

Ball B Northeast left English shot falls into the hole at count 6. Valid values for this shot are 3, 5, and 6.

Ball B Southeast right English shot is permitted to bounce to R9C3 from R9C4 because all rebounds are considered to happen in the center of a cell. For this specific shot, values of 7, 8, and 9 are not permitted because the value would be duplicated in the cell.

Possible paths for Ball C with a South shot. Notice that the English shot is the yellow line which makes a 45 degree angle between the blue line of the centered shot. Notice that 9 in not permitted on the centered shot because it would touch the same section of the wall twice.

The pink cells contain impossible shots. The 1 cannot be the digit for R6C1 because it would place multiple 1's in the box. Similarly R8C7 cannot be a 2 because although R6C9 seems valid, you cannot have 2 balls that have the digit 2. Also putting a 2 in R8C7 creates another 2 in the row and another 2 in the box.

A real bumper pool table!