At Inter-ACE 2018 we met an interesting programming challenge – Critters. The description read: “Evolve or die!” and the only file provided was an XPM image.

XPM itself is a rather odd format, where the image data is stored as a single valid C array in plaintext. I have no idea why that is, presumably so that source code can just #include <img.xpm>. But the challenge was not related to particularities of this format.

The provided image was a 64×64 black-and-white grid:

The content immediately made me think of Conway’s Game of Life, but I figured searching for the name might be worth it – and it turned out to be useful indeed. Critters is the name of a Life-like cellular automaton, but one which is reversible, unlike Life. This means that there is a unique mapping from states to their predecessors, and the initial state can be retrieved by running the simulation in reverse.

I was unable to find any existing code which would load arbitrary Critters worlds and advance them, so I wrote some myself using this informative website as reference. Critters is a block automaton where the world advances in 2×2 blocks, so I decided to use numpy due to its ability to easily work with submatrices of a matrix representing the world:

from PIL import Image
import numpy as np
import time


# Size in cells of the (square) world
DIM = 64

# Transcribed from XPM to a Python array using Vim search-and-replace
INIT_GRID = \
[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ],
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[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ] ]
INIT_GRID = np.array(INIT_GRID)

# Given an array of 4 elements representing a block in reading order,
# returns the block after one step
def step_block(block):
    num_alive =  sum(sum(block))
    if num_alive == 0:
        return np.ones(4).reshape(2, 2)
    elif num_alive == 1:
        return np.vectorize(lambda v: 1-v)(block)
    elif num_alive == 2:
        return block
    elif num_alive == 3:
        compl = np.vectorize(lambda v: 1-v)(block)
        return np.rot90(np.rot90(compl))
    elif num_alive == 4:
        return np.zeros(4).reshape(2, 2)

# Steps once and returns transformed grid
# pass_num - the number of the pass. whether even or odd matters
def step(grid, pass_num):
    if pass_num % 2 == 0: # even pass
        for i in range(0, DIM, 2):
            for j in range(0, DIM, 2):
                block = grid[i:i+2, j:j+2]
                block = step_block(block)
                grid[i:i+2, j:j+2] = block
        return grid
    else: # odd pass
        for i in range(1, DIM-1, 2):
            for j in range(1, DIM-1, 2):
                block = grid[i:i+2, j:j+2]
                block = step_block(block)
                grid[i:i+2, j:j+2] = block
        return grid

img = Image.new('RGB', (DIM, DIM), "white")
img_acc = img.load()

# Shows the world as ASCII art
def show(grid):
    for i in range(0, DIM):
        print(''.join([ '.' if c == 0 else 'H' for c in grid[0:DIM, i] ]))

# Shows the world as a PNG image
def show_img(grid, save=False,name="img.png"):
    for x in range(0, DIM):
        for y in range(0, DIM):
            val = grid[x, y] * 255
            img_acc[x, y] = (val, val, val)
    img.show()
    if save:
        img.save(name, "PNG")

def main():
    grid = INIT_GRID
    for i in range(0, 1000):
        grid = step(grid, i)
        print(i)
        show(grid)
        # The 9-th step is when we see the QR code
        if (i == 9):
            show_img(grid,True)
            break
            pass
        time.sleep(1)

if __name__ == "__main__":
    main()

As it turns out, running the simulation forwards for 9 steps results in a QR code containing the flag.

If we only needed to run it forwards, why did the authors decide to use a reversible automaton? One answer could be that they thought it would (and it did!) make the challenge more interesting than just finding a Game of Life simulator and plugging the data in.

Moreover, while it seems possible to reverse Game of Life in certain situations and the non-uniqueness of predecessor states is not a problem when we only care about the end state, it might be the case that for some manually drawn states there exists no predecessor. I was unable to find the answer to this, so either my Google-fu needs improvement or the problem would make for an interesting bit of new research.