The no plate of a bus had peculiarity. The bus number was a perfect square. It was also a perfect square when the plate was turned upside down. The bus company had only five hundred buses numbered from 1 to 500. What was the number?

Options

A. 169
B. 36
C. 196
D. Cannot say

Answer

To solve the problem, we need to find a bus number between 1 and 500 that is a perfect square and remains a perfect square when turned upside down. Here’s a step-by-step solution:

### Step 1: Identify the range of perfect squares
First, we need to find the perfect squares between 1 and 500. The largest integer whose square is less than or equal to 500 is 22, since \(22^2 = 484\). Therefore, we will consider perfect squares from \(1^2\) to \(22^2\).

### Step 2: List the perfect squares
The perfect squares from \(1^2\) to \(22^2\) are:
- \(1^2 = 1\)
- \(2^2 = 4\)
- \(3^2 = 9\)
- \(4^2 = 16\)
- \(5^2 = 25\)
- \(6^2 = 36\)
- \(7^2 = 49\)
- \(8^2 = 64\)
- \(9^2 = 81\)
- \(10^2 = 100\)
- \(11^2 = 121\)
- \(12^2 = 144\)
- \(13^2 = 169\)
- \(14^2 = 196\)
- \(15^2 = 225\)
- \(16^2 = 256\)
- \(17^2 = 289\)
- \(18^2 = 324\)
- \(19^2 = 361\)
- \(20^2 = 400\)
- \(21^2 = 441\)
- \(22^2 = 484\)

### Step 3: Identify valid digits for upside-down transformation
Next, we need to identify which digits can form valid numbers when turned upside down. The digits that can be transformed are:
- 0 → 0
- 1 → 1
- 6 → 9
- 8 → 8
- 9 → 6

### Step 4: Check which perfect squares can be turned upside down
Now we will check each perfect square to see if it consists only of the digits 0, 1, 6, 8, and 9, and if it remains a perfect square when turned upside down.

- \(1\) → remains \(1\) (perfect square)
- \(4\) → not valid
- \(9\) → remains \(9\) (perfect square)
- \(16\) → becomes \(91\) (not a perfect square)
- \(25\) → not valid
- \(36\) → becomes \(93\) (not a perfect square)
- \(49\) → not valid
- \(64\) → not valid
- \(81\) → becomes \(18\) (not a perfect square)
- \(100\) → becomes \(001\) (not a perfect square)
- \(121\) → remains \(121\) (perfect square)
- \(144\) → not valid
- \(169\) → becomes \(961\) (perfect square)
- \(196\) → becomes \(691\) (not a perfect square)
- \(225\) → not valid
- \(256\) → not valid
- \(289\) → not valid
- \(324\) → not valid
- \(361\) → becomes \(163\) (not a perfect square)
- \(400\) → becomes \(004\) (not a perfect square)
- \(441\) → not valid
- \(484\) → becomes \(484\) (perfect square)

### Step 5: Conclusion
The only perfect squares that remain perfect squares when turned upside down are \(1\), \(9\), \(121\), and \(169\). Among these, \(169\) turns into \(961\), which is also a perfect square.

Thus, the bus number that satisfies all the conditions is **196**.