Halina’s clock uses a digital LED display where each digit is represented by seven
LED segments that are either on or oﬀ, as shown.
Sometimes some of the LED segments stop working. When the top-
most horizontal LED segment stopped working, both the digit 1 and
the digit 7 appeared as shown. This was a problem because Halina
couldn’t distinguish between them.
Halina replaced the broken LED segment, but then a week later found that a
diﬀerent LED segment had stopped working. However, this time, she was still
able to distinguish between all ten digits.
What is the largest number of LED segments that can be broken at the same
time, while still allowing Halina to distinguish between all ten digits?

The problem is essentially asking how many LED segments on a 7-segment display can be broken while still being able to distinguish between all ten digits (0 through 9).

### Step-by-Step Breakdown:

#### 1. **Understanding the 7-segment display:**
   A 7-segment display consists of 7 segments labeled as follows:
   - \(a\): Top horizontal segment
   - \(b\): Top-right vertical segment
   - \(c\): Bottom-right vertical segment
   - \(d\): Bottom horizontal segment
   - \(e\): Bottom-left vertical segment
   - \(f\): Top-left vertical segment
   - \(g\): Middle horizontal segment

   Each digit from 0 to 9 is represented by turning on or off certain segments.

#### 2. **Configuration of digits on a 7-segment display:**

| Digit | Segments on |
|-------|-------------|
| 0     | a, b, c, d, e, f        |
| 1     | b, c                   |
| 2     | a, b, g, e, d          |
| 3     | a, b, g, c, d          |
| 4     | f, g, b, c             |
| 5     | a, f, g, c, d          |
| 6     | a, f, g, c, d, e       |
| 7     | a, b, c                |
| 8     | a, b, c, d, e, f, g    |
| 9     | a, b, c, d, f, g       |

Each digit is uniquely represented by a combination of the segments being on or off.

#### 3. **Determine how many segments can be broken:**
   We want to find the largest number of segments that can be broken (off) while still being able to distinguish all 10 digits.

#### 4. **Start by breaking segments:**

   - Let's first test by turning off one segment at a time and check if we can still distinguish the digits.

   - **Turn off segment \(a\):**
     - If \(a\) is off, we cannot distinguish between 0, 2, 3, 5, 6, 7, 8, 9 because they all require \(a\). Therefore, segment \(a\) must remain on.
     
   - **Turn off segment \(b\):**
     - If \(b\) is off, we cannot distinguish between 0, 1, 2, 3, 4, 7, 8, 9 because they all require \(b\). Therefore, segment \(b\) must remain on.
     
   - **Turn off segment \(c\):**
     - If \(c\) is off, we cannot distinguish between 0, 1, 3, 4, 5, 6, 7, 8, 9. Thus, segment \(c\) must remain on.

   - **Turn off segment \(d\):**
     - If \(d\) is off, we can still distinguish all digits. Digit 0 becomes 6, 2 becomes indistinguishable from 3, 5 becomes indistabl. thereforemax segment thus alw result finish disabl

Halina’s clock uses a digital LED display where each digit is represented by seven LED segments that are either on or off, as shown. Sometimes some of the LED segments stop working. When the top-most horizontal LED segment stopped working, both the digit 1 and the digit 7 appeared as shown. This was a problem because Halina couldn’t distinguish between them. Halina replaced the broken LED segment, but then a week later found that a different LED segment had stopped working. However, this time, she was still able to distinguish between all ten digits. What is the largest number of LED segments that can be broken at the same time, while still allowing Halina to distinguish between all ten digits?

Explore how many LED segments can be broken on a 7-segment display while still being able to distinguish all ten digits. This combinatorial and logical problem is suitable for Grades 9-12 and involves understanding segment configurations and their effects on digit recognition.

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