## Task 3 - 340623

A "number train" can be formed according to the following rules:

- In the first "car" stands a natural number greater than 1.
- If a "car" contains an even number, the next "car" contains half the number.
- If a "car" contains an odd number greater than 1, the next "car" contains the number reduced by 1.
- If a "car" contains the number 1, the "number train" ends with this "car".

a) Name all the "number trains" that consist of exactly 4 "cars"!

Justify that your list is complete!

b) What is the largest possible number that can appear in the first "car" of a "number train" that consists of exactly 7 "cars"?

Name such a "number train" with this starting number and show that a larger starting number is not possible!

c) What is the smallest possible number that can appear in the first "car" of a "number train" that consists of exactly 7 "cars"? Name such a "number train" with this starting number and show that a smaller starting number is not possible!
For each "number train":

In the last "carriage" stands a 1. If an odd number stands in a "carriage," then in the preceding "carriage" stands double that number. If an even number stands in a "carriage," then in the preceding "carriage" stands either the number that is one greater or double that number.

(a) From this, it follows in sequence:

In the second-to-last "carriage" stands a 2, before which stands either a 3 or a 4; before the 3 stands a 6; before the 4 stands either a 5 or an 8.

Thus, as a complete enumeration of "number trains" consisting of exactly 4 "carriages," it is justified:

$(6,3,2,1),(5,4,2,1),(8,4,2,1)$.

(b) If one chooses, for each even number, the option to take the preceding number as double, then one obtains as a "number train" consisting of exactly 7 "carriages": $(64,32,16,8,4,2,1)$.

Since doubling a number greater than 1 always results in a larger outcome than adding 1, a larger initial number is not possible; thus, 64 is proven to be the largest possible initial number.

(c) Following from (a), it follows in sequence:

As the fifth-to-last number, the only possibilities are: if it is odd, 7 and 9; if it is even, 10, 12, 16; as the sixth-to-last number: if it is odd, 11, 13, 17; if it is even, 14, 18, 20, 24, 32.

For the initial number preceding these, the smallest odd number is thus 15, the smallest even number is 22, and thus overall, 15 is proven to be the smallest possible initial number. The "number train" here is $(15,14,7,6,3,2,1)$.