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Palindromic Numbers

Eleven-Plus Preparation Specialists

This is something of a bonus piece for those interested in maths and scoring a high mark in the 11+. Palindromic numbers will sometimes feature as part of the test, normally being used as part of a puzzle question to differentiate good students from exceptional ones. As well as palindromic numbers, you should be aware of inverted numbers – which are numbers that look the same when reflected the top to bottom as opposed to the back to the front.

What is a Palindrome?

A palindrome is a number, word, sentence or even a full verse that is the same as it is back to front. A simple example is the name Anna. A more complex written example is the phrase, ‘A man, a plan, a canal, Panama.’ A palindromic number is just the same: a number that is the same one way as it is the other. The most obvious example that might come to mind is 121.

Let’s think about what that means in slightly more complex terms (as this is a guide for the high-achieving student): that means it has reflectional symmetry across the vertical axis. The first 30 palindromic numbers are:
0,1,2,3,4,5,6,7,8,9,11,22,33,44,55,66,77,88,99,101,121,131,141,151,161,171,181,191 and 202.

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How Can I Create a Palindromic Number?

It’s actually very simple to create a palindromic number, using low two digit numbers. All you need to do is write down any two ‘low’ two digit numbers. Then, write down the reverse of that two digit number. Then add these two numbers together. You will form a palindromic number. As an example, if we think about the number 121 again, then we would have used 74, reversed it to form 47, then added these together. Alternatively, let’s use the number 56. 56+65 = 121 as well! 43+34 = 77. However, if you go above 65 you will find it becomes much more difficult.

Using Larger Numbers

So, imagine that you have to use some larger numbers to create a palindrome. As before, we’ll begin by adding the original number to the reversed version. Let’s use the number 75.

75 + 57 = 132. However, 132 is clearly not a palindrome. Therefore, we need to reverse the digits of 132. 132 reversed gives us 231. Now, we need to add 132 to 231.

132 + 231 = 363. This is a palindrome! So, we can create palindromes from more complex or larger numbers too. Let’s look at another example.

The number 255 + 552 = 807. This is not a palindrome. Therefore, we add 807 to 708, which gives us 1515. 1515 + 5151 = 6666. This is a palindrome. So, with some extra steps we were able to make a palindrome again.

Inversions

Similar to palindromes are inversions. An inversion is a number that reads the same upside down as it does the right way up. You will often find these kinds of numbers brought up in exercises involving digital clocks, for example, where someone has realised that a number is the same one way up as it is the other. Let’s look at some examples:
96
689
830

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How many Palindromic Numbers are there?

Countless, of course. However, it’s worth knowing how many there are within each number of digits. There are 9 palindromic numbers with two digits, and there are 90 palindromic numbers with three digits (9 choices for the first digit, which means the third digit is the same, multiplied by 10 choices for the second digit).

There are also 9 palindromic numbers with four digits (9 choices for the first digit, and ten choices for the second digit. Then the other two digits have to be determined by the choices made for the first two).

Common Palindromes in Exam Questions

In very difficult 11+ tests, you could expect to be asked how many palindromes there could be between two particular numbers (use the rules above), or alternatively you might be asked to provide a palindromic date or time. A palindromic time might be 11:11, while a palindromic date could be the 22 of February 1922: or in other words, 22-2-22.
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