Digital Watch - Palindromic times

Probability Level 4

A 24-hour digital watch has many times that are palindromic. For example, 1:01:01, 2:41:42, 23:55:32, 3:59:53, 13:22:31, etc. (Ignore the colons.) How many palindromic combinations occur during a 24 hour?


The answer is 660.

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Ossama Ismail by Ossama Ismail , 63, Egypt

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1 solution

Marcelo Meneses
May 16, 2015

We divided the problem into three other:

  1. A:BC:BA ( i )
  2. 1A:BB:A1 ( ii )
  3. 2A:BB:A2 ( iii )

We must use the possibilities with more restrictions for calculating the multiplication

(i) 600 combinations 10 × 6 × 10 × 1 × 1 10 \times 6 \times 10 \times 1 \times 1

The "A" of hours can assume any integer value from 0 to 9: 10 combinations

The "B" of minutes can assume any integer value from 0 to 5: 6 combinations

The "C" of minutes can assume any integer value from 0 to 9: 10 combinations

The "B" of seconds can assume any integer value from 0 to 5: 6 combinations

The "A" of seconds can assume any integer value from 0 to 9: 10 combinations

The lowest number of possibilities for "A" is 10, to "B" is 6 and to "C" is 10, then only those numbers should be used in multiplication.

(ii) 36 combinations 1 × 1 × 6 × 1 × 6 × 1 1 \times 1 \times 6 \times 1 \times 6 \times 1

The "A" of hours can assume any integer value from 0 to 9: 10 combinations

The fist "B" of minutes can assume any integer value from 0 to 5: 6 combinations

The second "B" can assume any integer value from 0 to 9: 10 combinations

The "A" of seconds can assume any integer value from 0 to 5: 6 combinations

The lowest number of possibilities for "A" and "B" is 6 , then only those numbers should be used in multiplication.

(iii) 24 combinations 1 × 4 × 6 × 1 × 1 × 1 1 \times 4 \times 6 \times 1 \times 1 \times 1

The "A" of hours can assume any integer value from 0 to 3: 4 combinations

The fist "B" of minutes can assume any integer value from 0 to 5: 6 combinations

The second "B" can assume any integer value from 0 to 9: 10 combinations

The "A" of seconds can assume any integer value from 0 to 5: 6 combinations

The lowest number of possibilities for "A" is 4 and to "B" is 6 , then only those numbers should be used in multiplication.

So the answer is:

600 + 36 + 24 = 660 600+36+24=660

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Great and concise solution! I think this should be a combinatorics problem.

A Former Brilliant Member - 6 years ago

isnt 00:22:00 a palindrome

Shubham Garg - 6 years ago

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Yes, but 0:22:00 is not. This question says "For example, 1:01:01, 2:41:42, ..." So 00:22:00 is not included

Marcelo Meneses - 3 years, 5 months ago

sorry, yoy have counted the 24:00 hours(midnight) twice. You have considerated midnight in i) and in iii). Am I Wrong?

Guillermo Templado - 5 years, 9 months ago

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"(iii) 24 combinations The "A" of hours can assume any integer value from 0 to 3: 4 combinations"
24h is not included

24h => 0h

Marcelo Meneses - 3 years, 5 months ago

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