original source: [https://adventofcode.com/2021/day/11](https://adventofcode.com/2021/day/11)

You enter a large cavern full of rare bioluminescent [dumbo octopuses](https://www.youtube.com/watch?v=eih-VSaS2g0)! They seem to not like the Christmas lights on your submarine, so you turn them off for now.

There are 100 octopuses arranged neatly in a 10 by 10 grid. Each octopus slowly gains <em>energy</em> over time and <em>flashes</em> brightly for a moment when its energy is full. Although your lights are off, maybe you could navigate through the cave without disturbing the octopuses if you could predict when the flashes of light will happen.

Each octopus has an <em>energy level</em> - your submarine can remotely measure the energy level of each octopus (your puzzle input). For example:

<pre>

<code>5483143223

2745854711

5264556173

6141336146

6357385478

4167524645

2176841721

6882881134

4846848554

5283751526

</code>

</pre>

The energy level of each octopus is a value between <code>0</code> and <code>9</code>. Here, the top-left octopus has an energy level of <code>5</code>, the bottom-right one has an energy level of <code>6</code>, and so on.

You can model the energy levels and flashes of light in <em>steps</em>. During a single step, the following occurs:

- First, the energy level of each octopus increases by <code>1</code>.

- Then, any octopus with an energy level greater than <code>9</code> <em>flashes</em>. This increases the energy level of all adjacent octopuses by <code>1</code>, including octopuses that are diagonally adjacent. If this causes an octopus to have an energy level greater than <code>9</code>, it <em>also flashes</em>. This process continues as long as new octopuses keep having their energy level increased beyond <code>9</code>. (An octopus can only flash <em>at most once per step</em>.)

- Finally, any octopus that flashed during this step has its energy level set to <code>0</code>, as it used all of its energy to flash.

Adjacent flashes can cause an octopus to flash on a step even if it begins that step with very little energy. Consider the middle octopus with <code>1</code> energy in this situation:

<pre>

<code>Before any steps:

11111

19991

19191

19991

11111

After step 1:

34543

4<em>000</em>4

5<em>000</em>5

4<em>000</em>4

34543

After step 2:

45654

51115

61116

51115

45654

</code>

</pre>

An octopus is <em>highlighted</em> when it flashed during the given step.

Here is how the larger example above progresses:

<pre>

<code>Before any steps:

5483143223

2745854711

5264556173

6141336146

6357385478

4167524645

2176841721

6882881134

4846848554

5283751526

After step 1:

6594254334

3856965822

6375667284

7252447257

7468496589

5278635756

3287952832

7993992245

5957959665

6394862637

After step 2:

88<em>0</em>7476555

5<em>0</em>89<em>0</em>87<em>0</em>54

85978896<em>0</em>8

84857696<em>00</em>

87<em>00</em>9<em>0</em>88<em>00</em>

66<em>000</em>88989

68<em>0000</em>5943

<em>000000</em>7456

9<em>000000</em>876

87<em>0000</em>6848

After step 3:

<em>00</em>5<em>0</em>9<em>00</em>866

85<em>00</em>8<em>00</em>575

99<em>000000</em>39

97<em>000000</em>41

9935<em>0</em>8<em>00</em>63

77123<em>00000</em>

791125<em>000</em>9

221113<em>0000</em>

<em>0</em>421125<em>000</em>

<em>00</em>21119<em>000</em>

After step 4:

2263<em>0</em>31977

<em>0</em>923<em>0</em>31697

<em>00</em>3222115<em>0</em>

<em>00</em>41111163

<em>00</em>76191174

<em>00</em>53411122

<em>00</em>4236112<em>0</em>

5532241122

1532247211

113223<em>0</em>211

After step 5:

4484144<em>000</em>

2<em>0</em>44144<em>000</em>

2253333493

1152333274

11873<em>0</em>3285

1164633233

1153472231

6643352233

2643358322

2243341322

After step 6:

5595255111

3155255222

33644446<em>0</em>5

2263444496

2298414396

2275744344

2264583342

7754463344

3754469433

3354452433

After step 7:

67<em>0</em>7366222

4377366333

4475555827

34966557<em>0</em>9

35<em>00</em>6256<em>0</em>9

35<em>0</em>9955566

3486694453

8865585555

486558<em>0</em>644

4465574644

After step 8:

7818477333

5488477444

5697666949

46<em>0</em>876683<em>0</em>

473494673<em>0</em>

474<em>00</em>97688

69<em>0000</em>7564

<em>000000</em>9666

8<em>00000</em>4755

68<em>0000</em>7755

After step 9:

9<em>0</em>6<em>0000</em>644

78<em>00000</em>976

69<em>000000</em>8<em>0</em>

584<em>00000</em>82

5858<em>0000</em>93

69624<em>00000</em>

8<em>0</em>2125<em>000</em>9

222113<em>000</em>9

9111128<em>0</em>97

7911119976

After step 10:

<em>0</em>481112976

<em>00</em>31112<em>00</em>9

<em>00</em>411125<em>0</em>4

<em>00</em>811114<em>0</em>6

<em>00</em>991113<em>0</em>6

<em>00</em>93511233

<em>0</em>44236113<em>0</em>

553225235<em>0</em>

<em>0</em>53225<em>0</em>6<em>00</em>

<em>00</em>3224<em>0000</em>

</code>

</pre>

After step 10, there have been a total of <code>204</code> flashes. Fast forwarding, here is the same configuration every 10 steps:

<pre>

<code>After step 20:

3936556452

56865568<em>0</em>6

449655569<em>0</em>

444865558<em>0</em>

445686557<em>0</em>

568<em>00</em>86577

7<em>00000</em>9896

<em>0000000</em>344

6<em>000000</em>364

46<em>0000</em>9543

After step 30:

<em>0</em>643334118

4253334611

3374333458

2225333337

2229333338

2276733333

2754574565

5544458511

9444447111

7944446119

After step 40:

6211111981

<em>0</em>421111119

<em>00</em>42111115

<em>000</em>3111115

<em>000</em>3111116

<em>00</em>65611111

<em>0</em>532351111

3322234597

2222222976

2222222762

After step 50:

9655556447

48655568<em>0</em>5

448655569<em>0</em>

445865558<em>0</em>

457486557<em>0</em>

57<em>000</em>86566

6<em>00000</em>9887

8<em>000000</em>533

68<em>00000</em>633

568<em>0000</em>538

After step 60:

25333342<em>00</em>

274333464<em>0</em>

2264333458

2225333337

2225333338

2287833333

3854573455

1854458611

1175447111

1115446111

After step 70:

8211111164

<em>0</em>421111166

<em>00</em>42111114

<em>000</em>4211115

<em>0000</em>211116

<em>00</em>65611111

<em>0</em>532351111

7322235117

5722223475

4572222754

After step 80:

1755555697

59655556<em>0</em>9

448655568<em>0</em>

445865558<em>0</em>

457<em>0</em>86557<em>0</em>

57<em>000</em>86566

7<em>00000</em>8666

<em>0000000</em>99<em>0</em>

<em>0000000</em>8<em>00</em>

<em>0000000000</em>

After step 90:

7433333522

2643333522

2264333458

2226433337

2222433338

2287833333

2854573333

4854458333

3387779333

3333333333

After step 100:

<em>0</em>397666866

<em>0</em>749766918

<em>00</em>53976933

<em>000</em>4297822

<em>000</em>4229892

<em>00</em>53222877

<em>0</em>532222966

9322228966

7922286866

6789998766

</code>

</pre>

After 100 steps, there have been a total of <code><em>1656</em></code> flashes.

Given the starting energy levels of the dumbo octopuses in your cavern, simulate 100 steps. <em>How many total flashes are there after 100 steps?</em>

It seems like the individual flashes aren't bright enough to navigate. However, you might have a better option: the flashes seem to be <em>synchronizing</em>!

In the example above, the first time all octopuses flash simultaneously is step <code><em>195</em></code>:

<pre>

<code>After step 193:

5877777777

8877777777

7777777777

7777777777

7777777777

7777777777

7777777777

7777777777

7777777777

7777777777

After step 194:

6988888888

9988888888

8888888888

8888888888

8888888888

8888888888

8888888888

8888888888

8888888888

8888888888

After step 195:

<em>0000000000</em>

<em>0000000000</em>

<em>0000000000</em>

<em>0000000000</em>

<em>0000000000</em>

<em>0000000000</em>

<em>0000000000</em>

<em>0000000000</em>

<em>0000000000</em>

<em>0000000000</em>

</code>

</pre>

If you can calculate the exact moments when the octopuses will all flash simultaneously, you should be able to navigate through the cavern. <em>What is the first step during which all octopuses flash?</em>