Now let's look at a step-by-step example using a number that takes quite a while to reach the cycle: 27.

The rule is:

🟢 If it is even → divide by 2.

🔴 If it is odd → multiply by 3 and add 1.

Example: start with 27

Step 0:

[ 27 ]

27 is odd:

[ 27\times3+1=82 ]

Step 1:

[ 82 ]

82 is even:

[ 82\div2=41 ]

Step 2:

[ 41 ]

41 is odd:

[ 41\times3+1=124 ]

Step 3:

[ 124\div2=62 ]

Step 4:

[ 62\div2=31 ]

Step 5:
31 is odd:

[ 31\times3+1=94 ]

Step 6:

[ 94\div2=47 ]

Step 7:
47 is odd:

[ 47\times3+1=142 ]

Step 8:

[ 142\div2=71 ]

Step 9:
71 is odd:

[ 71\times3+1=214 ]

And it continues:

[ 214\rightarrow107\rightarrow322\rightarrow161\rightarrow484\rightarrow242\rightarrow121\rightarrow364\rightarrow182\rightarrow91\rightarrow274\rightarrow137\rightarrow412\righta rrow206\rightarrow103\rightarrow310\rightarrow155\rightarrow466\rightarrow233\rightarrow700\rightarrow350\rightarrow175\rightarrow526\rightarrow263\rightarrow790\rightarrow395\ri ghtarrow1186\rightarrow593\rightarrow1780\rightarrow890\rightarrow445\rightarrow1336\rightarrow668\rightarrow334\rightarrow167\rightarrow502\rightarrow251\rightarrow754\rightarr ow377\rightarrow1132\rightarrow566\rightarrow283\rightarrow850\rightarrow425\rightarrow1276\rightarrow638\rightarrow319\rightarrow958\rightarrow479\rightarrow1438\rightarrow719\r [ 2158\rightarrow1079\rightarrow3238\rightarrow1619\rightarrow4858\rightarrow2429\rightarrow7288\rightarrow3644\rightarrow1822\rightarrow911\rightarrow2734\rightarrow1367\rightarrow4102\rightarrow2051\rightarrow6154\rightarrow3077\rightarrow9232\rightarrow4616\rightarrow2308\rightarrow1154\rightarrow577\rightarrow1732\rightarrow866\rightarrow433\rightarrow1300\rightarrow650\rightarrow325\rightarrow976\rightarrow488\rightarrow244\rightarrow122\rightarrow61\rightarrow184\rightarrow92\rightarrow46\rightarrow23\rightarrow70\rightarrow35\rightarrow106\rightarrow53\rightarrow160\rightarrow80\rightarrow40\rightarrow20\rightarrow10\rightarrow5\rightarrow16\rightarrow8\rightarrow4\rightarrow2\rightarrow1 ]

And finally:

[ \boxed{1\rightarrow4\rightarrow2\rightarrow1} ]

The surprising part 🤯

The starting number was only 27, but before reaching 1, the sequence hits a maximum value of:

[ \boxed{9232} ]

In other words, it starts at 27, climbs to 9232, and eventually ends up at 1.

That behavior is precisely one of the reasons why the Collatz conjecture is so difficult: just because a number eventually goes down doesn't mean it decreases continuously.

And that brings up the big question:

Could there be a number that, unlike 27, never manages to return and enter the (4\rightarrow2\rightarrow1) loop?

So far, we have no way to prove that such a number doesn't exist.