Three Linked Fronts

Find all the first shots that guarantee victory, and demonstrate strategy against any adverse response.

Materials: Pencil and paper; optional chips, all pile numbers are given.

Rules you need

Original variant "pair subtraction". Each component is a pair of heaps A and B. The pairs remain independent.
At each turn, choose exactly ONE pair, then remove 1, 2 or 3 chips from A alone, OR B alone, OR exactly the same number (1, 2 or 3) of both piles. No pile can become negative. Never remove two different pairs in the same round.
The two sides alternate. The player without a legal blow loses: taking the last chips makes it win. There is neither pass nor rule misery. The tokens of the same pile are indistinguishable; the pairs and piles A/B are labeled.

The puzzle

You play first. To analyze alone, calculate the winning and losing positions of both sides.

Gives a complete list of the winners and a general proof of answer: only one cooperative suite is not enough.

Coordinates: lines from top to bottom, columns from left to right.
L / C	1	2	3
1	Pair	Pile A	Pile B
2	1	17	19
3	2	14	18
4	3	20	15

Why this level?

Build a Grundy table with two variables, assemble three components by XOR, list all the winning hits and prove the universal answer. The difficulty is on several combined tools, not on a long calculation of isolated chips. Editorial evaluation based on the required techniques; not calibrated by a human panel.

Your workspace

Write down your assumptions separately from deductions you have proved.

Notes for SOLO-039

View the solution to SOLO-039 →