Required Mathematical Intelligence

Chapter 57 Telephone Lines in Ukeley
It wasn't until last year that resistance to the phone was lifted in the Ukulele area.Although the installation of the telephone is now under way, the progress is relatively slow due to poor planning.

To this day, telephone lines between the six small towns in the area are far from complete. There are telephone lines between town A and the other five small towns; while towns B and C only have telephone lines with the other four small towns; towns D, E, and F only have telephone lines with the other three small towns. telephone line.If there is a complete telephone exchange system, the above phenomenon is not difficult to overcome.Because, if a telephone exchange system is installed in town A, the six towns of A, B, C, D, E, and F can communicate with each other.However, the telephone switching system will not be completed until half a year later.Prior to this, a direct line had to be installed between the two towns to communicate with each other.

Now, we also know that town D can call town F.

May I ask: Which three small towns can E town call?
[Answer: First of all, it is certain that there is a telephone line between town E and town A, because town A has telephone lines with the other five small towns.That of course includes Town E.

What are the other two small towns?

We start reasoning from the two small towns B and C.

Suppose: There is no telephone line between the two small towns B and C.Then, the two towns B and C must be able to communicate with the four towns A, D, E, and F respectively;
If the two towns B and C communicate with the four towns A, D, E and F respectively by phone, then the three towns D, E and F with only three telephone lines can only communicate with the three towns A, B and C respectively. make a call.

If so, then it is impossible to communicate on the phone between D, E, and F.

However, it is known that there is a telephone line between towns D and F, so the assumption that there is no telephone line between B and C cannot be established.In other words, there is a telephone line between the two small towns B and C.

So, which towns can towns B and C with four lines communicate with?

From the above reasoning, we know that Town B and Town C have telephone lines with Town A respectively, but there are no telephone lines between them.Where do the other two lines lead to?

Assumption: One of the other two lines in town B leads to town D and the other to town F; the telephone line in town C also has one to town D and the other to town F. If this assumption is true, then towns D and F will be connected There are four lines leading to other towns.However, we know that towns D and F only have telephone connections with three small towns, so the above assumption cannot be established.

Assumption: There are no telephone lines between towns B and C and towns D and F.

If this assumption is true, then towns B and C have only three lines to communicate with other towns, which does not meet the known condition that B and C each have four telephone lines.Therefore, the above assumption does not hold.

From the above analysis, it can only be deduced that towns B and C each have a telephone line leading to town E. Another route from town B either leads to town D or town F, and another route from town C either leads to town D or town F.

As for Town E, it can definitely communicate with the three towns A, B, and C. ]
(End of this chapter)