18. Show that if we allot 26 rooms to the students in a P.G hostel from the room numbered from 1 and 50
both inclusive at least two allotments are constitutively numbered.
19. A machine M is described by the state table below

Present State

Next State
Present Input Output z
x = 0 x = 1 x = 2
s0 s0 s1
s1

0

s1 s1 s0
s0

1

a. Draw the state graph for M.
b. What output corresponds to an input sequence of 2110?

### Problem 18: Prove that at least two rooms are consecutively numbered if 26 rooms are allotted from room numbers 1 to 50.

#### Solution:
This problem can be approached using the **Pigeonhole Principle**. The principle states that if more items are put into fewer containers than the number of items, at least one container must hold more than one item.

1. We are asked to allot 26 rooms from a total of 50 available rooms (numbered from 1 to 50).

2. Consider the worst-case scenario where no two rooms are consecutively numbered. In this case, each selected room must be separated by at least one other room.

3. If we select 26 rooms without any two being consecutive, the minimum room numbers for this selection would be:
   - Room 1, Room 3, Room 5, ..., Room 51

4. But Room 51 does not exist (since we have only rooms numbered up to 50). The highest room number is 50.

5. This shows that it is impossible to choose 26 rooms without selecting at least two consecutive room numbers. 

Thus, by the Pigeonhole Principle, when 26 rooms are allotted out of 50, there must be at least two rooms that are consecutively numbered.

### Problem 19: State Machine Analysis

Given the state table:

| Present State | Next State (x=0) | Next State (x=1) | Output z |
|---------------|------------------|------------------|----------|
| s0            | s0               | s1               | 0        |
| s1            | s1               | s0               | 1        |

#### a. Draw the State Graph for Machine M

To draw the state graph:
- The states are \( s0 \) and \( s1 \).
- The transitions are:
  - From \( s0 \) with input \( x = 0 \), the machine stays in \( s0 \) with output \( z = 0 \).
  - From \( s0 \) with input \( x = 1 \), the machine transitions to \( s1 \) with output \( z = 0 \).
  - From \( s1 \) with input \( x = 0 \), the machine stays in \( s1 \) with output \( z = 1 \).
  - From \( s1 \) with input \( x = 1 \), the machine transitions to \( s0 \) with output \( z = 1 \).

The state graph would consist of two nodes, \( s0 \) and \( s1 \), with directed edges representing the transitions according to the input values.

#### b. Determine the Output for Input Sequence 2110

Let’s analyze the machine’s behavior with the input sequence \( 2110 \):

1. **Start in \( s0 \):**
   - Input \( x = 2 \): Since \( x = 2 \) is not defined, assume that we ignore it or treat it as \( x = 0 \). So, stay in \( s0 \), output \( z = 0 \).
2. **Stay in \( s0 \) with \( x = 1 \):**
   - Transition to \( s1 \), output \( z = 0 \).
3. **Now in \( s1 \) with \( x = 1 \):**
   - Transition to \( s0 \), output \( z = 1 \).
4. **Now in \( s0 \) with \( x = 0 \):**
   - Stay in \( s0 \), output \( z = 0 \).

The output sequence corresponding to the input sequence \( 2110 \) is **0010**.

Would you like more details on any of these solutions?

Here are some related questions you might want to explore:

1. How can the Pigeonhole Principle be applied to more complex problems?
2. What happens if you change the number of rooms in Problem 18?
3. How would the output change if the state machine had more states?
4. What if the input sequence had more values (e.g., 3 or 4)?
5. How can we represent the state machine graphically using tools?

**Tip:** For state machines, drawing the state diagram first often clarifies how the machine transitions and what outputs to expect for given inputs.

Learn how to apply the Pigeonhole Principle to prove that at least two rooms are consecutively numbered when 26 rooms are allotted from numbers 1 to 50. Analyze state machine M with a state table to determine outputs for specific input sequences. Suitable for undergraduate level study.

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