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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.
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, â€˜A man, a plan, a canal, Panama.â€™ 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.
Letâ€™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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Itâ€™s actually very simple to create a palindromic number, using low two digit numbers. All you need to do is write down any two â€˜lowâ€™ 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, letâ€™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.
So, imagine that you have to use some larger numbers to create a palindrome. As before, weâ€™ll begin by adding the original number to the reversed version. Letâ€™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. Letâ€™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.
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. Letâ€™s look at some examples:
96
689
830
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