Visualization and solution to "seven nuts" puzzle. Requires DrawBot for the visualization.

SevenNutsPuzzle.py

# https://twitter.com/jenskutilek/status/1548996988892479488


def drawNuts(nuts, matches, nutPositions):
    for (bi1, bs1), (bi2, bs2) in matches:
        with savedState():
            strokeWidth(0.3 * distX)
            if nuts[bi1][bs1] == nuts[bi2][bs2]:
                stroke(0, 1, 0)
            else:
                stroke(1, 0, 0)
            line(nutPositions[bi1], nutPositions[bi2])

    for i, (x, y) in enumerate(nutPositions):
        drawNut(x, y, radius * 0.95, nuts[i])


def drawNut(cx, cy, r, numbers):
    with savedState():
        translate(cx, cy)
        points = [(r * cos(tau * i / 6), r * sin(tau * i / 6)) for i in range(6)]
        fill(0)
        polygon(*points)
        fill(1)
        drawCircle(0, 0, 0.35 * r)
        font("HelveticaNeue-CondensedBold")  # I don't have Franklin Gothic Condensed
        fontSize(0.35 * r)
        for i in range(6):
            a = tau * i / 6 - pi / 2
            text(str(numbers[i]), (0, -r / 1.3), align="center")
            rotate(60)


def drawCircle(cx, cy, r):
    d = r * 2
    oval(cx - r, cy - r, d, d)


def rotateNut(nut, i):
    i = i % 6
    if i:
        nut = nut[i:] + nut[:i]
    return nut


def isSolution(nuts, matches):
    for (bi1, bs1), (bi2, bs2) in matches:
        try:
            if nuts[bi1][bs1] != nuts[bi2][bs2]:
                return False
        except IndexError:
            # Partial check, ignore this combination
            pass
    return True


def solve(base, remaining, matches):
    if not remaining:
        yield base
        return
    for i in range(len(remaining)):
        first = remaining[i]
        rest = remaining[:i] + remaining[i + 1 :]
        for j in range(6):
            sol = base + [rotateNut(first, j)]
            if isSolution(sol, matches):
                yield from solve(sol, rest, matches)


# Nut configuration: start from center, then from bottom, go counter clockwise
#
#          nuts[4]
#   nuts[5]      nuts[3]
#          nuts[0]
#   nuts[6]      nuts[2]
#          nuts[1]
#

# Starting configuration as per photo
nuts = [
    [3, 4, 1, 2, 5, 6],  # counter clockwise from bottom
    [1, 5, 3, 2, 6, 4],
    [1, 2, 3, 4, 5, 6],
    [4, 2, 3, 5, 6, 1],
    [6, 1, 3, 5, 4, 2],
    [2, 4, 6, 1, 3, 5],
    [1, 6, 5, 4, 3, 2],
]

assert all(sorted(b) == [1, 2, 3, 4, 5, 6] for b in nuts)
assert all(len(set(b)) == 6 for b in nuts)
assert len({tuple(b) for b in nuts}) == 7


# Nut side indices 0..5 from BOTTOM side, counter-clockwise
matches = [
    # (nut index 1, nut side 1), (nut index 2, nut side 2)
    ((0, 0), (1, 3)),
    ((0, 1), (2, 4)),
    ((0, 2), (3, 5)),
    ((0, 3), (4, 0)),
    ((0, 4), (5, 1)),
    ((0, 5), (6, 2)),
    ((1, 2), (2, 5)),
    ((2, 3), (3, 0)),
    ((3, 4), (4, 1)),
    ((4, 5), (5, 2)),
    ((5, 0), (6, 3)),
    ((6, 1), (1, 4)),
]

assert len(matches) == 12

distY = 300
distX = distY * cos(pi / 6)
radius = distY / (2 * cos(pi / 6))

nutPositions = [
    (0, 0),
    (0, -distY),
    (distX, -distY / 2),
    (distX, distY / 2),
    (0, distY),
    (-distX, distY / 2),
    (-distX, -distY / 2),
]


solutions = list(solve([], nuts, matches))
print(f"there are {len(solutions)} solutions")
solution = solutions[0]

fill(1)
rect(0, 0, width(), height())
translate(500, 500)
drawNuts(solution, matches, nutPositions)
saveImage("SevenNutsPuzzle.png")