Questions on Palindromic & Sequence Numbers

In summary, There are 103 6-digit palindromic numbers and there are 900 odd 7-digit palindromic numbers where every digit appears at most twice. For the second question, the pigeon hole principle can be used to show that if 14 distinct integers are chosen from the sequence 100 to 124, there must be two of them whose difference is 4.
  • #1
chickenguy
16
0
more questions?

Hi more questions again... :blushing: :blushing: i am having a hard time figuring out these questions...
1)a)how many 6-digit palindromic numbers are there?
(i can do this the slow way, but i am looking for shortcuts)
b) how many odd 7-difit palindromic numbers are there in which every digit appears at most twice?
( again, i am looking for a fast way)

2)show that if 14 distinct intergers are chosen form the sequence 100, 101,102, 103...,123,124, there must be two of them whose difference is 4
(i have worked this out by writing out 14 numbers and having all the possibilities(it took a long time) and i am looking for a shortcut) :smile:
 
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  • #2
chickenguy said:
Hi more questions again... :blushing: :blushing: i am having a hard time figuring out these questions...
1)a)how many 6-digit palindromic numbers are there?
(i can do this the slow way, but i am looking for shortcuts)
What do you mean by the "slow way". Since the numbers are palindromic, once you have the first 3, the others must be the same. There are 103 such numbers.

b) how many odd 7-difit palindromic numbers are there in which every digit appears at most twice?
( again, i am looking for a fast way)
Lazy , eh? Since 7 is odd, we are looking for 4 digits. No digit more than twice? Okay, there are 10 possible first digits, then 10 possible second digits (which might be the same). We might want to separate those: there are 10(9)= 90 two different digit combos, 10 possible where the two digits are the same. Of the "different" two digits, there are again 10 possiblilities for the 3 digit but again we will want to consider separately that digit being the same or not the same as one of the first two... If the first two digits are the same then there are 9 possibilities for the third digit, etc.
I don't see any "easy" way. Sorry.

2)show that if 14 distinct intergers are chosen form the sequence 100, 101,102, 103...,123,124, there must be two of them whose difference is 4
(i have worked this out by writing out 14 numbers and having all the possibilities(it took a long time) and i am looking for a shortcut) :smile:
pigeon hole principal ought to work. How many different "differences" are possible?
 
  • #3
chickenguy said:
2)show that if 14 distinct intergers are chosen form the sequence 100, 101,102, 103...,123,124, there must be two of them whose difference is 4
(i have worked this out by writing out 14 numbers and having all the possibilities(it took a long time) and i am looking for a shortcut) :smile:

Hint: There's a reason the range is from N to N+24 (includes less than 14*2 - 1 integers)
 
  • #4
HallsofIvy said:
What do you mean by the "slow way". Since the numbers are palindromic, once you have the first 3, the others must be the same. There are 103 such numbers.

Actually, wouldn't it be 900, since the first digit can't be 0?
 

1. What is a palindrome number?

A palindrome number is a number that reads the same backward as it does forward. For example, 121 is a palindrome number because it reads the same from left to right and right to left.

2. How can I check if a number is a palindrome?

To check if a number is a palindrome, you can reverse the digits of the number and compare it to the original number. If the two numbers are the same, then the number is a palindrome. For example, if the number is 121, reversing the digits gives us 121, which is the same as the original number, so it is a palindrome.

3. What is a sequence number?

A sequence number is a number that follows a specific pattern or rule. It is a series of numbers that are generated by adding or multiplying a constant number to the previous number in the sequence. For example, the sequence 2, 4, 6, 8, 10 is generated by adding 2 to the previous number in the sequence.

4. How can I find the next number in a sequence?

To find the next number in a sequence, you need to identify the pattern or rule that the sequence is following. Once you have identified the pattern, you can use it to generate the next number in the sequence. For example, if the sequence is 3, 6, 9, 12, the pattern is multiplying each number by 3, so the next number in the sequence would be 15.

5. Can a number be both a palindrome and a sequence number?

Yes, a number can be both a palindrome and a sequence number. For example, the number 121 is a palindrome as well as a sequence number. It follows the pattern of adding 11 to the previous number in the sequence (11, 22, 33, 44, etc.).

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