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The Hailstone sequence of numbers can be generated from a starting positive integer,   n   by:

* If   n   is     **1**     then the sequence ends.
* If   n   is   **even** then the next   n   of the sequence    `= n/2`
* If   n   is   **odd**   then the next   n   of the sequence    `= (3 * n) + 1`

The (unproven) Collatz conjecture is that the hailstone sequence for any starting number always terminates.

This sequence was named by Lothar Collatz in 1937   (or possibly in 1939),   and is also known as (the):

:   :   :   * hailstone sequence,   hailstone numbers
            * 3x + 2 mapping,   3n + 1 problem
            * Collatz sequence
            * Hasse's algorithm
            * Kakutani's problem
            * Syracuse algorithm,   Syracuse problem
            * Thwaites conjecture
            * Ulam's problem

The hailstone sequence is also known as   *hailstone numbers*   (because the values are usually subject to multiple descents and ascents like hailstones in a cloud).

Task

1. Create a routine to generate the hailstone sequence for a number.
2. Use the routine to show that the hailstone sequence for the number 27 has 112 elements starting with `27, 82, 41, 124` and ending with `8, 4, 2, 1`
3. Show the number less than 100,000 which has the longest hailstone sequence together with that sequence's length.  
      (But don't show the actual sequence!)

See also

* xkcd (humourous).
* The Notorious Collatz conjecture Terence Tao, UCLA (Presentation, pdf).
* The Simplest Math Problem No One Can Solve Veritasium (video, sponsored).
assistant
(formerly Perl 6)

```
sub hailstone($n) { $n, { $_ %% 2 ?? $_ div 2 !! $_ * 3 + 1 } ... 1 }

my @h = hailstone(27);
say "Length of hailstone(27) = {+@h}";
say ~@h;

my $m = max ( (1..99_999).race.map: { +hailstone($_) => $_ } );
say "Max length {$m.key} was found for hailstone({$m.value}) for numbers < 100_000";
```

Output:

```
Length of hailstone(27) = 112
27 82 41 124 62 31 94 47 142 71 214 107 322 161 484 242 121 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
Max length 351 was found for hailstone(77031) for numbers < 100_000

```