Logic Masters Deutschland e.V.

Puzzle link: Play on SudokuPad.

Short rules: Normal Sudoku rules apply. The red lines are electric wires, the gray rectangles resistors (with their resistance as sum over digits), the circles current meters, the red points junctions, and the yellow-marked cells batteries (with a voltage written between the black rectangular poles, for two-cell batteries written as a two-digit number with tens in the cell of the longer positive pole and ones in the cell of the shorter negative pole). All circuits have to fulfill Ohm's law (i.e., the voltage drop over a resistor is the resistance times the current), Kirchhoff's junction rule (i.e., the currents add up to zero at a junction when taking into account their directions towards or away from the junction with plus or minus signs, respectively), and Kirchhoff's loop rule (i.e., the voltages add up to zero over every loop when taking into account the correct directions of voltages by appropriate plus or minus signs, specifically such that battery voltages compensate voltage drops over resistors). Digits have to increase in the direction of the technical current on each uninterrupted sequence of cells along a wire if that sequence does not contain any current meters, batteries, resistors, or junctions.

Long rules: Normal Sudoku rules apply.

A “wire” is an uninterrupted and unbranching red line. A wire transports electric “current”, i.e., the value of the current is the same everywhere along a wire.

A “current meter” is a circle that displays the absolute magnitude of the current in the wires to which it is attached. The current must be the same in both attached wires. While the current in a wire can be an arbitrary real number, its absolute magnitude must be an integer and a valid Sudoku digit if it is displayed by a current meter.

A “battery” is a yellow region of one or two neighboring cells. It contains a “positive pole” (long, thin black rectangle) and a “negative pole” (short, thick black rectangle). The “voltage” of the battery is given by the number it contains. For a two-cell battery, the voltage is written as a two-digit number with the tens digit in the cell of the positive pole and the ones digit in the cell of the negative pole. The current must be the same in the two wires attached to the positive and the negative pole.

A “resistor” is a gray rectangle with a “resistance” given by the sum of its digits. The current must be the same in both attached wires.

A “junction” is a red dot that connects three or more wires.

A “loop” is an uninterrupted closed path along wires that may additionally pass through current meters, batteries, resistors, and/or junctions such that it starts from an arbitrary point on a wire in either of the two directions of the wire and returns to the same point from the other direction without crossing itself anywhere along the path.

An “electric circuit” is a complete set of all wires, current meters, batteries, resistors, and/or junctions that are connected to each other through wires.

For each loop of an electric circuit, assume a specific but arbitrary loop direction (e.g., clockwise or counterclockwise) along all wires and, if present, through the current meters, batteries, resistors, and/or junctions that the loop contains. If a wire is part of more than one loop, the assumed loop directions do not have to agree with each other for that wire.

For each uninterrupted path from one particular junction to the next junction that this path reaches in an electric circuit, assume a specific but arbitrary current direction. The path may pass along wires and, if present, through current meters, batteries, and/or resistors. If an electric circuit contains no junctions, assume a current direction that is the same throughout the whole loop. Assumed current directions do not have to agree with assumed loop directions.

According to Ohm's law, a resistor introduces a voltage between the two connected wires that is given by its resistance times the current. Neither a wire nor a current meter introduce voltages on their own.

According to Kirchhoff's first law (“junction rule”), the “algebraic sum” is zero of all currents through all wires connected at a junction. Calculating the algebraic sum means that a current is added to the sum if its assumed direction points towards the junction and subtracted otherwise.

According to Kirchhoff's second law (“loop rule”), the algebraic sum is zero of all voltages around a loop. Calculating the algebraic sum means that the voltage of a battery is added to the sum if the assumed loop direction points from the positive pole to the negative pole within the battery and subtracted otherwise; the voltage of a resistor is added to the sum if the assumed current direction through the resistor agrees with the assumed loop direction and subtracted otherwise.

Each individual value for a current fulfilling Kirchhoff's laws can in principle be positive, zero, or negative. A positive number means that the “technical current” direction agrees with the assumed current direction, a negative number means that the technical current direction is opposite to the assumed current direction, and zero means that there is no technical current.

Ohm's and Kirchhoff's laws must be fulfilled for all parts of all electric circuits.

Digits have to increase in the direction of the technical current on each uninterrupted sequence of cells along a wire if that sequence does not contain any current meters, batteries, resistors, or junctions; single-cell sequences are not restricted in value.

Your feedback, ratings and comments are highly appreciated. Have fun!

Example: You can play the 4x4 grid below yourself on SudokuPad. Note that three loops can be defined: a top loop, a bottom loop, and an outer loop all the way around. Kirchhoff's second law is fulfilled for all three. In the path along rows 1 and 2, the current is 1 and the resistance is 3; in the path along row 3, the current is 3 and the resistance is 1. Both the outer loop and the bottom loop thus require a voltage of 3 according to each individual product of resistance times current, which is provided by the battery. The technical current direction is from left to right in rows 1 and 2, from left to right in row 3, and from right to left in row 4. The currents at the junction add up according to 1+3=4.