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The Binary Watch Trivia Quiz

Browsing through a duty-free brochure on a recent flight, I spotted an ad for a binary watch. While it was a bit expensive, can you afford to refuse a free quiz?

A multiple-choice quiz by gentlegiant17. Estimated time: 23 mins.

Author
gentlegiant17

Time
23 mins

Type
Multiple Choice

Quiz #
306,815

Updated
Dec 03 21

# Qns
10

Difficulty
Difficult

Avg Score
4 / 10

Plays
2475

Awards
Top 5% quiz!

Question 1 of 10

1. My binary watch has two rows of LEDs (Light Emitting Diodes).
On its top row, a certain number of LEDs for telling hours.
On its bottom row, a certain numbers of LEDs for telling minutes.

For example, when displaying the hour 03:07, 5 LEDs in total are turned ON:
The 2 rightmost ones of the hour row (as the binary representation of 3 is 11) plus the 3 rightmost ones of the minute row (as the binary representation of 7 is 111).

My watch does not have an AM/PM indicator so please consider hours to be between 0 and 23. Minutes are between 0 and 59. This version of the watch does NOT tell seconds.

In order to be able to display all times between 00:00 and 23:59, how many LEDs in total does my watch have?

Answer: (an odd two-digit number, please answer in decimal)

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Question 2 of 10

2. In which of the following times of the day NO two adjacent LEDs are turned ON (i.e. adjacent in a given row)? Hint

10:42

10:43

11:42

11:43

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Question 3 of 10

3. In which of the following times of the day is the largest number of LEDs turned ON? Hint

07:27

19:43

12:09

15:47

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Question 4 of 10

4. What is the maximum number of LEDs in my watch which can be turned ON at a given time of the day?

Answer: (one digit, please answer in decimal)

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Question 5 of 10

5. The leftmost hour and the leftmost minute LEDs are BOTH turned ON.

How many times a day does this happen?

Answer: (a three-digit number, please answer in decimal)

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Question 6 of 10

6. As a single minute passes, what is the maximum number of LEDs which can switch from ON to OFF?

Answer: (one digit, please answer in decimal)

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Question 7 of 10

7. Assuming that the only factor which determines the lifetime of a LED in my watch is the number of times it shifts between ON and OFF states and that all LEDs were installed at the same time, which is the LED that will last the longest time? Hint

The leftmost minute LED

The rightmost hour LED

The rightmost minute LED

The leftmost hour LED

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Question 8 of 10

8. What time is it when ONLY the 4 external LEDs in my watch are turned ON (i.e. leftmost and rightmost hour LEDs, leftmost and rightmost minute LEDs)?

Answer: (a time of the day in HH:MM format, e.g. 03:07)

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Question 9 of 10

9. Which is the earliest time of the day when 7 LEDs in total are turned ON?

Answer: (a time of the day in HH:MM format, e.g. 03:07)

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Question 10 of 10

10. This last question may look short and sweet, but be warned not to lose yourself when performing the calculation - it's tricky.

How many times a day exactly 3 LEDs in my watch are turned ON?

Answer: (an even three-digit number, please answer in decimal)

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Quiz Answer Key and Fun Facts

1. My binary watch has two rows of LEDs (Light Emitting Diodes). On its top row, a certain number of LEDs for telling hours. On its bottom row, a certain numbers of LEDs for telling minutes. For example, when displaying the hour 03:07, 5 LEDs in total are turned ON: The 2 rightmost ones of the hour row (as the binary representation of 3 is 11) plus the 3 rightmost ones of the minute row (as the binary representation of 7 is 111). My watch does not have an AM/PM indicator so please consider hours to be between 0 and 23. Minutes are between 0 and 59. This version of the watch does NOT tell seconds. In order to be able to display all times between 00:00 and 23:59, how many LEDs in total does my watch have?

Answer: 11

My watch has 5 LEDs for showing hours: 23 lies between 16 (10000 in binary) and 32 (100000 in binary).
It has 6 LEDs for showing minutes: 59 lies between 32 (100000 in binary) and 64 (1000000 in binary).
Thus, 11 LEDs in total are required.

2. In which of the following times of the day NO two adjacent LEDs are turned ON (i.e. adjacent in a given row)?

Answer: 10:42

Below, "X" signifies a LED which is OFF and "O" signifies a LED which is ON.

You don't want two adjacent O's in your correct answer.

10:42 looks like this:

XOXOX
OXOXOX

10:43 looks like this:

XOXOX
OXOXOO

11:42 looks like this:

XOXOO
OXOXOX

11:43 looks like this:

XOXOO
OXOXOO

3. In which of the following times of the day is the largest number of LEDs turned ON?

Answer: 15:47

Again, "X" signifies a LED which is OFF and "O" signifies a LED which is ON.

07:27 (7 LEDs on):

XXOOO
XOOXOO

12:09 (4 LEDs on):

XOOXX
XXOXXO

15:47 (9 LEDs on):

XOOOO
OXOOOO

19:43 (7 LEDs on):

OXXOO
OXOXOO

4. What is the maximum number of LEDs in my watch which can be turned ON at a given time of the day?

Answer: 9

The watch can never have all of its hour LEDs on (since 11111 binary is 31 decimal and there is no such hour of the day).

The watch can never have all of its minute LEDs on (since 111111 binary is 63 decimal and there is no such minute of the hour).

There is a number of times during the day where 4 hour LEDs and 5 minute LEDs are on, yielding a maximum total of 9 lit LEDs (hmmmm... can you tell how many?).

5. The leftmost hour and the leftmost minute LEDs are BOTH turned ON. How many times a day does this happen?

Answer: 224

The leftmost hour LED is ON at all hours between 16 and 23 (8 in total).
The leftmost minute LED is ON at all minutes between 32 and 59 (28 in total).
Thus, both these LEDs are ON 224 times a day (8*28=224).

6. As a single minute passes, what is the maximum number of LEDs which can switch from ON to OFF?

Answer: 9

Yes, there is an occasion where the number of lit LEDs decreases from its maximum value of 9 to its minimum value of 0 - the passage from 23:59 to midnight.

Answer: The leftmost hour LED

The good old leftmost hour LED shifts between states only twice daily: OFF to ON at 16:00 and ON to OFF at 00:00. It is by far the least used LED in my watch.

8. What time is it when ONLY the 4 external LEDs in my watch are turned ON (i.e. leftmost and rightmost hour LEDs, leftmost and rightmost minute LEDs)?

Answer: 17:33

The binary representation of 17 is 10001.
The binary representation of 33 is 100001.

Thus, at 17:33 the LED scheme of my watch is the desired one:

OXXXO
OXXXXO

Where "X" signifies a LED which is OFF and "O" signifies a LED which is ON.

9. Which is the earliest time of the day when 7 LEDs in total are turned ON?

Answer: 03:31

Since it is impossible for all the 6 minute LEDs to be simultaneously on, you are looking for the earliest time where 2 hour LEDs and 5 minute LEDs are on. The meaning of "earliest" is that these LEDs should be justified to the right.

Representing the hour 03:31, here is the LED scheme in question:

XXXOO
XOOOOO

Where "X" signifies a LED which is OFF and "O" signifies a LED which is ON.

10. This last question may look short and sweet, but be warned not to lose yourself when performing the calculation - it's tricky. How many times a day exactly 3 LEDs in my watch are turned ON?

Answer: 156

This one requires a bit of thought and flexibility which will enable you to consider the combinatorics along with the algebra of the watch.

A plain combinatorical calculation of choosing 3 of 11 LEDs will give an answer of 165 (11!/3!*8!). However, this is not the correct answer (which is why I hinted you that I am looking for an even number). Consider the 9 combinations where the 2 leftmost hour LEDs are on - all of them are illegal as they produce an hour of the day larger than 23 (in other words, 11000 binary is 24 decimal).

Thus, the correct answer is 156 (165 combinations minus 9 illegal ones).

Source: Author gentlegiant17

This quiz was reviewed by FunTrivia editor crisw before going online.
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