Why does a scrambled image come back?

Each rule only moves pixels around, so every pixel travels in a loop. When the step count is a multiple of every loop length at once (their least common multiple), all pixels are home and the picture returns exactly.

How many zips does it take for a 1024×1024 image to come back?

10. A perfect out-shuffle of 1024 rows sends row r to row 2r modulo 1023, and 2^10 = 1024 is 1 more than 1023. A 1022×1022 image needs 340 zips, a 1000×1000 image needs 36.

Is my image uploaded anywhere?

No. Everything runs in your browser. The picture never leaves your device.

Will It Come Back? Scramble Any Image With Math and Watch It Return

Why does a scrambled picture come back?

Every rule here only moves pixels. No pixel is created, destroyed or recolored. So if you follow one pixel, it hops from spot to spot and, because there are only so many spots, it must eventually land where it started. Every pixel travels in its own loop.

The whole picture is back when every loop finishes at the same moment. That happens at the least common multiple of all the loop lengths. This page finds every loop in your picture and computes that number exactly, even when it has hundreds of digits.

The zip: 1024 comes back in 10, 1022 needs 340

A zip is a perfect shuffle of rows, then of columns. With 1024 rows, row r moves to row 2r (counting modulo 1023). After 10 zips every row has been doubled 10 times, and 2¹⁰ = 1024 is exactly 1 more than 1023, so everyone is home. Halfway there, at 5 zips, the picture becomes 1,024 tiny copies of itself.

Take two rows away and the story changes completely. 1022 needs 340 zips, 1000 needs 36, and 1026 needs 20. Try the size box and the table under it: the number of steps jumps around wildly between neighbours.

Arnold's cat map and the baker's map

The cat map shears the square and wraps it around like a torus. The baker's map squashes, stretches and stacks, like kneading dough. Both are textbook examples of chaos: nearby pixels fly apart fast and the picture turns to noise within a few steps. Yet because the grid is finite, both must come back. A 101×101 cat comes back after 25 steps. A 1000×1000 knead takes about 2.13 × 10²⁵³.

Ring twist

Every square ring moves one pixel clockwise. The outer ring of a 64×64 picture has 252 pixels, the next 244, and so on. Each ring is home after its own length, but all of them together need the least common multiple: about 7.38 × 10²⁵ steps.

Rearrange: the same pixels, a different picture

Pick a target and watch every pixel of your picture fly to a new place. Dark pixels go where the target is dark, bright ones where it is bright, then nearby pairs swap if that brings both colors closer. It is still the very same set of pixels, so it can always fly back.

Send a secret picture

Scramble a photo to step k and save the PNG. Anyone who knows the rule, the size and k can turn it back: open it with “Use exact pixels”, then step backward k times. It is a fun puzzle, not real encryption.

Watch the videos

Made by Computed on YouTube.

ぐちゃぐちゃの絵は、なぜ元に戻る？

ここにある規則はどれも画素を動かすだけです。新しく作る画素も、消す画素も、色を変える画素もありません。1 つの画素を追いかけると、席から席へ跳んでいき、席の数は限られているので、いつか必ず出発した席に戻ってきます。どの画素も、自分の輪を回っています。

絵全体が戻るのは、すべての輪が同時に 1 周し終えたときです。それは輪の長さの最小公倍数の手数です。このページは絵の中の輪をすべて見つけ、その数を正確に計算します。何百桁になってもです。

ジップ: 1024 は 10 回、1022 は 340 回

ジップは行の完全シャッフル、次に列の完全シャッフルです。1024 行なら r 行目は 2r 行目（1023 で割った余り）へ移ります。10 回で全員が 10 回 2 倍され、2¹⁰ = 1024 は 1023 よりちょうど 1 大きいので、全員が元の席に戻ります。半分の 5 回目では、絵は自分の小さな複製 1,024 個になります。

2 行減らすだけで話はまったく変わります。1022 は 340 回、1000 は 36 回、1026 は 20 回。大きさの欄と、その下の表で試してください。隣どうしの大きさでも手数は大きく飛びます。

アーノルドの猫写像とパン屋の変換

猫写像は正方形を斜めにずらし、ドーナツのように端をつなげます。パン屋の変換はつぶして伸ばして重ねる、生地をこねる動きです。どちらもカオスの教科書的な例で、近い画素はすぐに離れ、数手で絵は砂嵐になります。それでもマス目が有限なので、必ず戻ります。101×101 の猫は 25 手、1000×1000 のこねは約 2.13 × 10²⁵³ 手です。

輪の回転

四角い輪がそれぞれ時計回りに 1 マス進みます。64×64 の一番外の輪は 252 画素、次は 244 画素…。輪はそれぞれ自分の長さで戻りますが、全部そろうのは最小公倍数、約 7.38 × 10²⁵ 手です。

並べ替え: 同じ画素で、別の絵

目標の絵を選ぶと、あなたの絵の画素がすべて新しい場所へ飛びます。暗い画素は目標の暗い所へ、明るい画素は明るい所へ。そのあと近くの 2 つを入れ替えて色が近づくなら入れ替えます。同じ画素の集まりのままなので、いつでも飛んで戻れます。

秘密の絵を送る

写真を k 手目までぐちゃぐちゃにして PNG を保存します。規則・大きさ・k を知っている人なら戻せます。「画素をそのまま使う」で開き、k 手戻すだけです。本物の暗号ではなく、遊びのパズルです。

動画で見る

制作: YouTube の Computed