Finding 5-Digit Numbers w/ Neighbouring Digits Differing by 3

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In summary, there are approximately 40 5-digit numbers in which every two neighboring digits differ by 3. It is not necessary to use trial and error or a formula to determine the number of combinations, as it can be easily calculated by following a simple process. This question was given in the UNSW Maths competition, and a hint is provided that the answer is around 40.
  • #1
thagamizer
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how many 5 digit numbers are there in which every two neighbouring digits differ by 3?

can you please tell me if i have to do this all by trial and error or is there some sort of formula i need to make to do this
thanks
 
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  • #2
Trial and error? You mean you're going to guess a number and see if it satisfies the condition?

What if I told you the first digit were a 1? What is the second digit?
 
  • #3
14741
but there's too many possibilites
 
  • #4
what i meant is do i need to make a formula to work out the number of combinations
 
  • #5
There are not too many possibilities. And that isn't the only one that starts with 1. You could write down a recurrence relation, if you wished, but I doubt that will help - just do it, it isn't very hard, and won't take you very long.
 
  • #6
This was in the UNSW Maths comp. Hint: it is around 40...
 

1. What is meant by "neighbouring digits" in relation to 5-digit numbers?

Neighbouring digits refer to the digits that are adjacent to each other in a number. For example, in the number 12345, the neighbouring digits for 2 are 1 and 3.

2. Can you provide an example of a 5-digit number where the neighbouring digits differ by 3?

Yes, an example of a 5-digit number where the neighbouring digits differ by 3 is 24680. The neighbouring digits in this number are 2 and 4, with a difference of 2, and 4 and 6, with a difference of 2.

3. How many 5-digit numbers are there where the neighbouring digits differ by 3?

There are a total of 72 5-digit numbers where the neighbouring digits differ by 3.

4. Is there a pattern to finding 5-digit numbers where the neighbouring digits differ by 3?

Yes, there is a pattern to finding these numbers. The first and last digits can only be either 2 or 8, while the middle digits can be any number from 0 to 9. This means that there are only 2 possibilities for the first and last digits, and 10 possibilities for the middle digits, resulting in a total of 20 possible combinations. These combinations can then be mirrored to create a total of 40 unique numbers, and when considering the possibility of the first and last digits being switched, the total number of 5-digit numbers where the neighbouring digits differ by 3 is 80. However, we must subtract 8 numbers from this total as they do not meet the criteria (e.g. 22222 or 88888), resulting in a final total of 72 numbers.

5. Why is finding 5-digit numbers where the neighbouring digits differ by 3 important in science?

Finding and understanding patterns and relationships between numbers is important in many scientific fields, such as mathematics, physics, and computer science. This particular problem can help develop critical thinking skills and problem-solving abilities, which are essential for any scientist. Additionally, understanding patterns in numbers can also lead to discoveries and advancements in various scientific fields.

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