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Elim Puzzle

You are given a square board (4×4, 6×6, or 8×8) with numbers as tokens. The sum of the absolute value of all the tokens is the token value count, which is shown on the left.
Select a row by clicking one of the arrows. The arrows for the other rows change to plus and minus signs. Click one of those signs at the end of another row . Then is replaced by or , according to the sign you clicked. Columns act in a similar way. Zero is not displayed.
The goal is to reduce the token value count as much as possible. Even for the smallest board (4×4) the puzzle is surprisingly hard to solve.

SNAPSHOTS

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DETAILS

The author found the target amounts for setups 1, 2, and 3 of boards 4×4 and 6×6 by hand, so you may be able to improve on them.
The best target amounts for the three 8×8 setups are not known (see also the description of the control "setup" below).
Controls
"board size": Selects the size of the board (4×4, 6×6, or 8×8).
"setup": There are three challenges with fixed setups for each board size. The fourth setup has an empty board. Click one of the corner positions to randomize the setup, or click board positions repeatedly to manually set up the board. Once you start moves you can no longer change the setup.
"colored": Selects whether you want the tokens colored or white only.
"move": Displays the number of the current move and also the total amount of moves.
"<<", "< 10", "<" and ">", "> 10", ">>": These two setter bars let you select previous moves and so on.
"repeat move 1× / 10×": Repeats the last move once or ten times.
"save", "restore": Saves or restores the current sequence of moves.
"token value count": Shows the sum of the absolute values of all tokens on the board. Try to make this sum as small as possible.
"target": Shows the smallest token count the author has found. Can you get this count or even improve on it?
"solution": Solutions are stored only for the setups 1 to 3, which use the boards 4×4 and 6×6. Use the paging controls ("<<", "<", ">", ">>") to go through a solution move by move.
Variants
The Elim puzzle allows for other goals as well. For example, you can try to end with the smallest number of tokens on the board, independent of their value.
For the 6×6 board, this means that you try to end up with six tokens (or maybe even fewer). The third fixed setup for the 6×6 type allows such an ending with six tokens. Can you find it?
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