Lever Law (6x6)
(Published on 25. August 2024, 10:39 by Tobias Brixner)
Puzzle link: Play on SudokuPad.
Background: The lever law is already known since ancient times and is sometimes stated as: “The product of the lever arm and the force is the same for both the effort and the load.” With a sufficiently long lever arm, large loads can therefore be moved with small force applied. Archimedes is often quoted as saying: “Give me a lever long enough and a fulcrum on which to place it, and I shall move the world.” (“Fulcrum” is another word for “pivot”.) The lever law can be described physically using torques. A lever is in equilibrium when no torque acts on it. The underlying mathematics are realized in this puzzle.
Rules: Normal 6x6 Sudoku rules apply.
A red line symbolizes a lever with a pivot marked by the white dot it contains. A part of the lever between two vertical grid lines is a lever segment. A stack consists of the digits above a lever segment. Stacks end at the next lever above, at a small white rectangle, or at the top of the grid, whichever comes first. The weight of a stack is equal to the sum of its digits.
Consider the weight of a stack on a lever segment multiplied by the distance of the segment from the pivot. A lever is balanced if summing up these products for all segments left of its pivot is the same as summing up all these products right of its pivot. All levers must be balanced. A stack directly over a pivot has no influence. The distance of a lever segment is measured in numbers of cells from the middle of the segment to the pivot. If the pivot is halfway between vertical grid lines, distances are whole numbers; if it is on a vertical grid line, distances are half numbers.
Example: The following image provides a fully solved example on a 4x4 grid. You may solve the example for yourself here on SudokuPad.
Solution code: All digits of row 6 (from left to right) followed by column 3 (from top to bottom) without spaces.
Comments
on 26. August 2024, 15:44 by brimmy
I like this idea using physics ( mechanic)
on 26. August 2024, 14:19 by brimmy
I like this idea using physics ( mechanic)
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That's great to hear, thanks for playing! If you enjoy physics in Sudoku, I have published several other puzzles with connections to physics that you can find when clicking on my name above. Examples are "Uncertainty Principle" (000HXV), "Laser Maze" (000HRX), "Definition of Temperature" (000HOM), "Physical Constants" (000HSV), or "Schrödinger Equation" (000HS8). - TB
on 25. August 2024, 13:53 by GorgeousNicko
Took a while to get my head around the rules. After that, very smooth. As for the halves, I found it simplest to double everything. In this case that just meant adding 3 x the more distant figure. If, as I hope, there is a 9x9 sequel, we could get into 5s and 7s!
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Thanks a lot for commenting and motivating me to attempt a sequel. I have a couple of other Sudoku projects I want to finish first, but maybe afterwards... Indeed, the employed distance "units" are irrelevant if you use the same on both sides of the pivot. - TB
on 25. August 2024, 13:46 by RockyRoer
The levers with 0.5 lengths was a little challenging, until I started thinking about them as halves - multiplying by 1/2, or 3/2 and comparing halves.
Fun little math exercise! Thanks for setting and sharing it!
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Thank you so much for playing and commenting, glad you liked it! - TB
