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  41 Oh, hello!  Funny seeing you here.
  42
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 153
 154 <main>
 155 <article class=3D"day-desc"><h2>--- Day 9: Rope Bridge ---</h2><p>This rope=
 156  bridge creaks as you walk along it. You aren't sure how old it is, or whet=
 157 her it can even support your weight.</p>
 158 <p>It seems to support the Elves just fine, though. The bridge spans a gorg=
 159 e which was carved out by the massive river far below you.</p>
 160 <p>You step carefully; as you do, the ropes stretch and twist. You decide t=
 161 o distract yourself by modeling rope physics; maybe you can even figure out=
 162  where <em>not</em> to step.</p>
 163 <p>Consider a rope with a knot at each end; these knots mark the <em>head</=
 164 em> and the <em>tail</em> of the rope. If the head moves far enough away fr=
 165 om the tail, the tail is pulled toward the head.</p>
 166 <p>Due to nebulous reasoning involving <a href=3D"https://en.wikipedia.org/=
 167 wiki/Planck_units#Planck_length" target=3D"_blank" rel=3D"noreferrer" data-=
 168 ss1670926033=3D"1">Planck lengths</a>, you should be able to model the posi=
 169 tions of the knots on a two-dimensional grid. Then, by following a hypothet=
 170 ical <em>series of motions</em> (your puzzle input) for the head, you can d=
 171 etermine how the tail will move.</p>
 172 <p><span title=3D"I'm an engineer, not a physicist!">Due to the aforementio=
 173 ned Planck lengths</span>, the rope must be quite short; in fact, the head =
 174 (<code>H</code>) and tail (<code>T</code>) must <em>always be touching</em>=
 175  (diagonally adjacent and even overlapping both count as touching):</p>
 176 <pre><code>....
 177 .TH.
 178 ....
 179
 180 ....
 181 .H..
 182 ..T.
 183 ....
 184
 185 ...
 186 .H. (H covers T)
 187 ...
 188 </code></pre>
 189 <p>If the head is ever two steps directly up, down, left, or right from the=
 190  tail, the tail must also move one step in that direction so it remains clo=
 191 se enough:</p>
 192 <pre><code>.....    .....    .....
 193 .TH.. -&gt; .T.H. -&gt; ..TH.
 194 .....    .....    .....
 195
 196 ...    ...    ...
 197 .T.    .T.    ...
 198 .H. -&gt; ... -&gt; .T.
 199 ...    .H.    .H.
 200 ...    ...    ...
 201 </code></pre>
 202 <p>Otherwise, if the head and tail aren't touching and aren't in the same r=
 203 ow or column, the tail always moves one step diagonally to keep up:</p>
 204 <pre><code>.....    .....    .....
 205 .....    ..H..    ..H..
 206 ..H.. -&gt; ..... -&gt; ..T..
 207 .T...    .T...    .....
 208 .....    .....    .....
 209
 210 .....    .....    .....
 211 .....    .....    .....
 212 ..H.. -&gt; ...H. -&gt; ..TH.
 213 .T...    .T...    .....
 214 .....    .....    .....
 215 </code></pre>
 216 <p>You just need to work out where the tail goes as the head follows a seri=
 217 es of motions. Assume the head and the tail both start at the same position=
 218 , overlapping.</p>
 219 <p>For example:</p>
 220 <pre><code>R 4
 221 U 4
 222 L 3
 223 D 1
 224 R 4
 225 D 1
 226 L 5
 227 R 2
 228 </code></pre>
 229 <p>This series of motions moves the head <em>right</em> four steps, then <e=
 230 m>up</em> four steps, then <em>left</em> three steps, then <em>down</em> on=
 231 e step, and so on. After each step, you'll need to update the position of t=
 232 he tail if the step means the head is no longer adjacent to the tail. Visua=
 233 lly, these motions occur as follows (<code>s</code> marks the starting posi=
 234 tion as a reference point):</p>
 235 <pre><code>=3D=3D Initial State =3D=3D
 236
 237 ......
 238 ......
 239 ......
 240 ......
 241 H.....  (H covers T, s)
 242
 243 =3D=3D R 4 =3D=3D
 244
 245 ......
 246 ......
 247 ......
 248 ......
 249 TH....  (T covers s)
 250
 251 ......
 252 ......
 253 ......
 254 ......
 255 sTH...
 256
 257 ......
 258 ......
 259 ......
 260 ......
 261 s.TH..
 262
 263 ......
 264 ......
 265 ......
 266 ......
 267 s..TH.
 268
 269 =3D=3D U 4 =3D=3D
 270
 271 ......
 272 ......
 273 ......
 274 ....H.
 275 s..T..
 276
 277 ......
 278 ......
 279 ....H.
 280 ....T.
 281 s.....
 282
 283 ......
 284 ....H.
 285 ....T.
 286 ......
 287 s.....
 288
 289 ....H.
 290 ....T.
 291 ......
 292 ......
 293 s.....
 294
 295 =3D=3D L 3 =3D=3D
 296
 297 ...H..
 298 ....T.
 299 ......
 300 ......
 301 s.....
 302
 303 ..HT..
 304 ......
 305 ......
 306 ......
 307 s.....
 308
 309 .HT...
 310 ......
 311 ......
 312 ......
 313 s.....
 314
 315 =3D=3D D 1 =3D=3D
 316
 317 ..T...
 318 .H....
 319 ......
 320 ......
 321 s.....
 322
 323 =3D=3D R 4 =3D=3D
 324
 325 ..T...
 326 ..H...
 327 ......
 328 ......
 329 s.....
 330
 331 ..T...
 332 ...H..
 333 ......
 334 ......
 335 s.....
 336
 337 ......
 338 ...TH.
 339 ......
 340 ......
 341 s.....
 342
 343 ......
 344 ....TH
 345 ......
 346 ......
 347 s.....
 348
 349 =3D=3D D 1 =3D=3D
 350
 351 ......
 352 ....T.
 353 .....H
 354 ......
 355 s.....
 356
 357 =3D=3D L 5 =3D=3D
 358
 359 ......
 360 ....T.
 361 ....H.
 362 ......
 363 s.....
 364
 365 ......
 366 ....T.
 367 ...H..
 368 ......
 369 s.....
 370
 371 ......
 372 ......
 373 ..HT..
 374 ......
 375 s.....
 376
 377 ......
 378 ......
 379 .HT...
 380 ......
 381 s.....
 382
 383 ......
 384 ......
 385 HT....
 386 ......
 387 s.....
 388
 389 =3D=3D R 2 =3D=3D
 390
 391 ......
 392 ......
 393 .H....  (H covers T)
 394 ......
 395 s.....
 396
 397 ......
 398 ......
 399 .TH...
 400 ......
 401 s.....
 402 </code></pre>
 403 <p>After simulating the rope, you can count up all of the positions the <em=
 404 >tail visited at least once</em>. In this diagram, <code>s</code> again mar=
 405 ks the starting position (which the tail also visited) and <code>#</code> m=
 406 arks other positions the tail visited:</p>
 407 <pre><code>..##..
 408 ...##.
 409 .####.
 410 ....#.
 411 s###..
 412 </code></pre>
 413 <p>So, there are <code><em>13</em></code> positions the tail visited at lea=
 414 st once.</p>
 415 <p>Simulate your complete hypothetical series of motions. <em>How many posi=
 416 tions does the tail of the rope visit at least once?</em></p>
 417 </article>
 418 <p>Your puzzle answer was <code>6243</code>.</p><article class=3D"day-desc"=
 419 ><h2 id=3D"part2">--- Part Two ---</h2><p>A rope snaps! Suddenly, the river=
 420  is getting a lot closer than you remember. The bridge is still there, but =
 421 some of the ropes that broke are now whipping toward you as you fall throug=
 422 h the air!</p>
 423 <p>The ropes are moving too quickly to grab; you only have a few seconds to=
 424  choose how to arch your body to avoid being hit. Fortunately, your simulat=
 425 ion can be extended to support longer ropes.</p>
 426 <p>Rather than two knots, you now must simulate a rope consisting of <em>te=
 427 n</em> knots. One knot is still the head of the rope and moves according to=
 428  the series of motions. Each knot further down the rope follows the knot in=
 429  front of it using the same rules as before.</p>
 430 <p>Using the same series of motions as the above example, but with the knot=
 431 s marked <code>H</code>, <code>1</code>, <code>2</code>, ..., <code>9</code=
 432 >, the motions now occur as follows:</p>
 433 <pre><code>=3D=3D Initial State =3D=3D
 434
 435 ......
 436 ......
 437 ......
 438 ......
 439 H.....  (H covers 1, 2, 3, 4, 5, 6, 7, 8, 9, s)
 440
 441 =3D=3D R 4 =3D=3D
 442
 443 ......
 444 ......
 445 ......
 446 ......
 447 1H....  (1 covers 2, 3, 4, 5, 6, 7, 8, 9, s)
 448
 449 ......
 450 ......
 451 ......
 452 ......
 453 21H...  (2 covers 3, 4, 5, 6, 7, 8, 9, s)
 454
 455 ......
 456 ......
 457 ......
 458 ......
 459 321H..  (3 covers 4, 5, 6, 7, 8, 9, s)
 460
 461 ......
 462 ......
 463 ......
 464 ......
 465 4321H.  (4 covers 5, 6, 7, 8, 9, s)
 466
 467 =3D=3D U 4 =3D=3D
 468
 469 ......
 470 ......
 471 ......
 472 ....H.
 473 4321..  (4 covers 5, 6, 7, 8, 9, s)
 474
 475 ......
 476 ......
 477 ....H.
 478 .4321.
 479 5.....  (5 covers 6, 7, 8, 9, s)
 480
 481 ......
 482 ....H.
 483 ....1.
 484 .432..
 485 5.....  (5 covers 6, 7, 8, 9, s)
 486
 487 ....H.
 488 ....1.
 489 ..432.
 490 .5....
 491 6.....  (6 covers 7, 8, 9, s)
 492
 493 =3D=3D L 3 =3D=3D
 494
 495 ...H..
 496 ....1.
 497 ..432.
 498 .5....
 499 6.....  (6 covers 7, 8, 9, s)
 500
 501 ..H1..
 502 ...2..
 503 ..43..
 504 .5....
 505 6.....  (6 covers 7, 8, 9, s)
 506
 507 .H1...
 508 ...2..
 509 ..43..
 510 .5....
 511 6.....  (6 covers 7, 8, 9, s)
 512
 513 =3D=3D D 1 =3D=3D
 514
 515 ..1...
 516 .H.2..
 517 ..43..
 518 .5....
 519 6.....  (6 covers 7, 8, 9, s)
 520
 521 =3D=3D R 4 =3D=3D
 522
 523 ..1...
 524 ..H2..
 525 ..43..
 526 .5....
 527 6.....  (6 covers 7, 8, 9, s)
 528
 529 ..1...
 530 ...H..  (H covers 2)
 531 ..43..
 532 .5....
 533 6.....  (6 covers 7, 8, 9, s)
 534
 535 ......
 536 ...1H.  (1 covers 2)
 537 ..43..
 538 .5....
 539 6.....  (6 covers 7, 8, 9, s)
 540
 541 ......
 542 ...21H
 543 ..43..
 544 .5....
 545 6.....  (6 covers 7, 8, 9, s)
 546
 547 =3D=3D D 1 =3D=3D
 548
 549 ......
 550 ...21.
 551 ..43.H
 552 .5....
 553 6.....  (6 covers 7, 8, 9, s)
 554
 555 =3D=3D L 5 =3D=3D
 556
 557 ......
 558 ...21.
 559 ..43H.
 560 .5....
 561 6.....  (6 covers 7, 8, 9, s)
 562
 563 ......
 564 ...21.
 565 ..4H..  (H covers 3)
 566 .5....
 567 6.....  (6 covers 7, 8, 9, s)
 568
 569 ......
 570 ...2..
 571 ..H1..  (H covers 4; 1 covers 3)
 572 .5....
 573 6.....  (6 covers 7, 8, 9, s)
 574
 575 ......
 576 ...2..
 577 .H13..  (1 covers 4)
 578 .5....
 579 6.....  (6 covers 7, 8, 9, s)
 580
 581 ......
 582 ......
 583 H123..  (2 covers 4)
 584 .5....
 585 6.....  (6 covers 7, 8, 9, s)
 586
 587 =3D=3D R 2 =3D=3D
 588
 589 ......
 590 ......
 591 .H23..  (H covers 1; 2 covers 4)
 592 .5....
 593 6.....  (6 covers 7, 8, 9, s)
 594
 595 ......
 596 ......
 597 .1H3..  (H covers 2, 4)
 598 .5....
 599 6.....  (6 covers 7, 8, 9, s)
 600 </code></pre>
 601 <p>Now, you need to keep track of the positions the new tail, <code>9</code=
 602 >, visits. In this example, the tail never moves, and so it only visits <co=
 603 de><em>1</em></code> position. However, <em>be careful</em>: more types of =
 604 motion are possible than before, so you might want to visually compare your=
 605  simulated rope to the one above.</p>
 606 <p>Here's a larger example:</p>
 607 <pre><code>R 5
 608 U 8
 609 L 8
 610 D 3
 611 R 17
 612 D 10
 613 L 25
 614 U 20
 615 </code></pre>
 616 <p>These motions occur as follows (individual steps are not shown):</p>
 617 <pre><code>=3D=3D Initial State =3D=3D
 618
 619 ..........................
 620 ..........................
 621 ..........................
 622 ..........................
 623 ..........................
 624 ..........................
 625 ..........................
 626 ..........................
 627 ..........................
 628 ..........................
 629 ..........................
 630 ..........................
 631 ..........................
 632 ..........................
 633 ..........................
 634 ...........H..............  (H covers 1, 2, 3, 4, 5, 6, 7, 8, 9, s)
 635 ..........................
 636 ..........................
 637 ..........................
 638 ..........................
 639 ..........................
 640
 641 =3D=3D R 5 =3D=3D
 642
 643 ..........................
 644 ..........................
 645 ..........................
 646 ..........................
 647 ..........................
 648 ..........................
 649 ..........................
 650 ..........................
 651 ..........................
 652 ..........................
 653 ..........................
 654 ..........................
 655 ..........................
 656 ..........................
 657 ..........................
 658 ...........54321H.........  (5 covers 6, 7, 8, 9, s)
 659 ..........................
 660 ..........................
 661 ..........................
 662 ..........................
 663 ..........................
 664
 665 =3D=3D U 8 =3D=3D
 666
 667 ..........................
 668 ..........................
 669 ..........................
 670 ..........................
 671 ..........................
 672 ..........................
 673 ..........................
 674 ................H.........
 675 ................1.........
 676 ................2.........
 677 ................3.........
 678 ...............54.........
 679 ..............6...........
 680 .............7............
 681 ............8.............
 682 ...........9..............  (9 covers s)
 683 ..........................
 684 ..........................
 685 ..........................
 686 ..........................
 687 ..........................
 688
 689 =3D=3D L 8 =3D=3D
 690
 691 ..........................
 692 ..........................
 693 ..........................
 694 ..........................
 695 ..........................
 696 ..........................
 697 ..........................
 698 ........H1234.............
 699 ............5.............
 700 ............6.............
 701 ............7.............
 702 ............8.............
 703 ............9.............
 704 ..........................
 705 ..........................
 706 ...........s..............
 707 ..........................
 708 ..........................
 709 ..........................
 710 ..........................
 711 ..........................
 712
 713 =3D=3D D 3 =3D=3D
 714
 715 ..........................
 716 ..........................
 717 ..........................
 718 ..........................
 719 ..........................
 720 ..........................
 721 ..........................
 722 ..........................
 723 .........2345.............
 724 ........1...6.............
 725 ........H...7.............
 726 ............8.............
 727 ............9.............
 728 ..........................
 729 ..........................
 730 ...........s..............
 731 ..........................
 732 ..........................
 733 ..........................
 734 ..........................
 735 ..........................
 736
 737 =3D=3D R 17 =3D=3D
 738
 739 ..........................
 740 ..........................
 741 ..........................
 742 ..........................
 743 ..........................
 744 ..........................
 745 ..........................
 746 ..........................
 747 ..........................
 748 ..........................
 749 ................987654321H
 750 ..........................
 751 ..........................
 752 ..........................
 753 ..........................
 754 ...........s..............
 755 ..........................
 756 ..........................
 757 ..........................
 758 ..........................
 759 ..........................
 760
 761 =3D=3D D 10 =3D=3D
 762
 763 ..........................
 764 ..........................
 765 ..........................
 766 ..........................
 767 ..........................
 768 ..........................
 769 ..........................
 770 ..........................
 771 ..........................
 772 ..........................
 773 ..........................
 774 ..........................
 775 ..........................
 776 ..........................
 777 ..........................
 778 ...........s.........98765
 779 .........................4
 780 .........................3
 781 .........................2
 782 .........................1
 783 .........................H
 784
 785 =3D=3D L 25 =3D=3D
 786
 787 ..........................
 788 ..........................
 789 ..........................
 790 ..........................
 791 ..........................
 792 ..........................
 793 ..........................
 794 ..........................
 795 ..........................
 796 ..........................
 797 ..........................
 798 ..........................
 799 ..........................
 800 ..........................
 801 ..........................
 802 ...........s..............
 803 ..........................
 804 ..........................
 805 ..........................
 806 ..........................
 807 H123456789................
 808
 809 =3D=3D U 20 =3D=3D
 810
 811 H.........................
 812 1.........................
 813 2.........................
 814 3.........................
 815 4.........................
 816 5.........................
 817 6.........................
 818 7.........................
 819 8.........................
 820 9.........................
 821 ..........................
 822 ..........................
 823 ..........................
 824 ..........................
 825 ..........................
 826 ...........s..............
 827 ..........................
 828 ..........................
 829 ..........................
 830 ..........................
 831 ..........................
 832
 833 </code></pre>
 834 <p>Now, the tail (<code>9</code>) visits <code><em>36</em></code> positions=
 835  (including <code>s</code>) at least once:</p>
 836 <pre><code>..........................
 837 ..........................
 838 ..........................
 839 ..........................
 840 ..........................
 841 ..........................
 842 ..........................
 843 ..........................
 844 ..........................
 845 #.........................
 846 #.............###.........
 847 #............#...#........
 848 .#..........#.....#.......
 849 ..#..........#.....#......
 850 ...#........#.......#.....
 851 ....#......s.........#....
 852 .....#..............#.....
 853 ......#............#......
 854 .......#..........#.......
 855 ........#........#........
 856 .........########.........
 857 </code></pre>
 858 <p>Simulate your complete series of motions on a larger rope with ten knots=
 859 . <em>How many positions does the tail of the rope visit at least once?</em=
 860 ></p>
 861 </article>
 862 <p>Your puzzle answer was <code>2630</code>.</p><p class=3D"day-success">Bo=
 863 th parts of this puzzle are complete! They provide two gold stars: **</p>
 864 <p>At this point, you should <a href=3D"https://adventofcode.com/2022" data=
 865 -ss1670926033=3D"1">return to your Advent calendar</a> and try another puzz=
 866 le.</p>
 867 <p>If you still want to see it, you can <a href=3D"https://adventofcode.com=
 868 /2022/day/9/input" target=3D"_blank" data-ss1670926033=3D"1">get your puzzl=
 869 e input</a>.</p>
 870 <p>You can also <span class=3D"share">[Share<span class=3D"share-content">o=
 871 n
 872   <a href=3D"https://twitter.com/intent/tweet?text=3DI%27ve+completed+%22Ro=
 873 pe+Bridge%22+%2D+Day+9+%2D+Advent+of+Code+2022&amp;url=3Dhttps%3A%2F%2Fadve=
 874 ntofcode%2Ecom%2F2022%2Fday%2F9&amp;related=3Dericwastl&amp;hashtags=3DAdve=
 875 ntOfCode" target=3D"_blank" rel=3D"noreferrer" data-ss1670926033=3D"1">Twit=
 876 ter</a>
 877   <a target=3D"_blank" rel=3D"noreferrer" data-ss1670926033=3D"1">Mastodon<=
 878 /a></span>]</span> this puzzle.</p>
 879 </main>
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2574 none; cursor: default; }
2575
2576 .calendar a:hover, .calendar a:focus { background-color: rgba(119, 119, 165=
2577 , 0.2); cursor: pointer; }
2578
2579 #calendar-countdown { padding-left: 0.5em; color: rgb(204, 204, 204); }
2580
2581 .calendar .calendar-day { color: rgb(102, 102, 102); }
2582
2583 .calendar a .calendar-day { color: rgb(204, 204, 204); }
2584
2585 .calendar a .calendar-mark-complete, .calendar a .calendar-mark-verycomplet=
2586 e { visibility: hidden; }
2587
2588 .calendar a.calendar-complete .calendar-mark-complete, .calendar a.calendar=
2589 -verycomplete .calendar-mark-complete { visibility: visible; color: rgb(255=
2590 , 255, 102); }
2591
2592 .calendar a.calendar-verycomplete .calendar-mark-verycomplete { visibility:=
2593  visible; color: rgb(255, 255, 102); }
2594
2595 .calendar .calendar-day-new { animation: 5s ease 0s 1 normal none running a=
2596 nim-day-new; }
2597
2598 .calendar .calendar-day-new .calendar-day { animation: 5s ease 0s 1 normal =
2599 none running anim-day-new-day; }
2600
2601 @keyframes anim-day-new {=20
2602   0% { color: rgb(51, 51, 51); text-shadow: transparent 0px 0px 5px; }
2603   25% { color: rgb(255, 255, 255); text-shadow: rgb(255, 255, 255) 0px 0px =
2604 5px; }
2605   100% { color: rgb(102, 102, 102); text-shadow: transparent 0px 0px 5px; }
2606 }
2607
2608 @keyframes anim-day-new-day {=20
2609   0% { color: rgb(102, 102, 102); text-shadow: transparent 0px 0px 5px; }
2610   25% { color: rgb(255, 255, 255); text-shadow: rgb(255, 255, 255) 0px 0px =
2611 5px; }
2612   100% { color: rgb(204, 204, 204); text-shadow: transparent 0px 0px 5px; }
2613 }
2614
2615 .eventlist-event { white-space: pre; }
2616
2617 .stats > span, .stats > span .stats-firstonly, .stats > span .stats-both { =
2618 color: rgb(102, 102, 102); }
2619
2620 .stats > a { color: rgb(204, 204, 204); min-width: 35em; display: inline-bl=
2621 ock; }
2622
2623 .stats > a:hover, .stats > a:focus { background-color: rgb(30, 30, 70); }
2624
2625 .stats-firstonly { color: rgb(153, 153, 204); }
2626
2627 .stats-both { color: rgb(255, 255, 102); }
2628
2629 .leaderboard-daylinks { cursor: default; }
2630
2631 .leaderboard-daylinks-selected { color: rgb(255, 255, 255); text-shadow: rg=
2632 b(255, 255, 255) 0px 0px 5px; }
2633
2634 .leaderboard-daydesc-first { color: rgb(153, 153, 204); }
2635
2636 .leaderboard-daydesc-both { color: rgb(255, 255, 102); }
2637
2638 .leaderboard-entry { white-space: pre; }
2639
2640 .leaderboard-entry .leaderboard-totalscore { color: rgb(255, 255, 255); }
2641
2642 .leaderboard-anon { opacity: 0.6; }
2643
2644 .leaderboard-userphoto { display: inline-block; height: 20px; width: 20px; =
2645 margin: 0px 0.5em; text-align: center; }
2646
2647 .leaderboard-userphoto img { height: 20px; max-width: 20px; vertical-align:=
2648  middle; position: relative; top: -2px; }
2649
2650 .leaderboard-time { opacity: 0.5; }
2651
2652 .privboard-row { white-space: pre; }
2653
2654 .privboard-name { vertical-align: text-bottom; }
2655
2656 .privboard-days > span { display: inline-block; color: rgb(51, 51, 51); }
2657
2658 .privboard-days > a { display: inline-block; }
2659
2660 .privboard-star-locked { visibility: hidden; }
2661
2662 .privboard-star-unlocked { color: rgb(51, 51, 51); }
2663
2664 .privboard-star-firstonly { color: rgb(153, 153, 204); }
2665
2666 .privboard-star-both { color: rgb(255, 255, 102); }
2667
2668 .privboard-delbtn { opacity: 0.33; }
2669
2670 .privboard-row:hover .privboard-delbtn { opacity: 1; }
2671
2672 .sponsors { width: 46em; }
2673
2674 .sponsor { margin: 1em 0px; }
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2676 Content-Type: text/css
2677 Content-Transfer-Encoding: quoted-printable
2678 Content-Location: https://adventofcode.com/static/highcontrast.css?0
2679
2680 @charset "utf-8";
2681
2682 * { background: white !important; color: black !important; text-shadow: non=
2683 e !important; }
2684
2685 a { color: rgb(0, 0, 255) !important; text-decoration: underline !important=
2686 ; }
2687
2688 em { font-weight: 900 !important; }
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