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?
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
New Palindromic Numbers Questions and Answers
1. What is a palindromic number? Give three examples and explain what they have in common.
A palindromic number reads the same forwards as it does backwards. Three examples are 121, 1331, and 45654. What they have in common is that the digit sequence is symmetrical around the centre: the first digit matches the last, the second matches the second-to-last, and so on. Single-digit numbers (1 through 9) are technically all palindromes because they read the same in either direction, but in 11+ questions the term usually refers to numbers with two or more digits. The two-digit palindromes are 11, 22, 33, 44, 55, 66, 77, 88, and 99, all of which share the property of being divisible by 11. Recognising palindromic numbers quickly is a useful skill in 11+ puzzle questions, where they often appear as part of a multi-step problem designed to separate the strongest candidates from the rest.
2. A three-digit palindromic number has 7 as its middle digit. The sum of all three digits is 17. What are the two possible numbers?
The two possible numbers are 585 and 494. A three-digit palindrome has the form ABA, where the first and last digits are the same. The sum of digits is A plus 7 plus A = 2A plus 7 = 17, so 2A = 10 and A = 5. That gives only one number: 575. Let me recheck: 2A + 7 = 17 means A = 5, giving 575. There is only one answer: 575. Checking: 5 + 7 + 5 = 17 and it reads the same forwards and backwards. Three-digit palindrome problems are solved most reliably by setting up the ABA form first, writing the digit sum as an equation, and solving for A. The structure of the palindrome does all the work, reducing a puzzle that might seem complex to a single straightforward equation.
3. How many palindromic numbers are there between 100 and 999 inclusive? Explain your reasoning.
There are 90 three-digit palindromic numbers between 100 and 999. A three-digit palindrome has the form ABA, where A is the first and last digit and B is the middle digit. A can be any digit from 1 to 9 (it cannot be 0, as that would make it a two-digit number), giving 9 choices. B can be any digit from 0 to 9, giving 10 choices. Since A and the last digit are the same, there is no further choice once A and B are selected. Total palindromes = 9 x 10 = 90. Counting questions like this reward systematic reasoning over listing examples. Identifying the structure of the palindrome (ABA) and counting the independent choices for each position is far faster and more reliable than writing out every possible number, and it is the method examiners expect to see at this level.
4. Take any two-digit number, reverse its digits, and add the two numbers together. Show that the result is always a palindrome for the number 73.
73 reversed is 37. Adding: 73 plus 37 = 110. That is not a palindrome. Let me recheck with 63: 63 plus 36 = 99, which is a palindrome. The rule that reversing and adding always produces a palindrome holds for most two-digit numbers but not all: numbers where the digit sum is 10 or more (like 73, where 7 + 3 = 10) can produce results that require a second iteration. 110 reversed is 011 = 11, and 110 plus 11 = 121, which is a palindrome. So for 73, one iteration is not enough: two are needed. This is a known property of palindromic addition, and 11+ puzzle questions sometimes test whether candidates notice that the process occasionally needs repeating. The correct answer is that the result is always eventually a palindrome, but for some starting numbers it takes more than one step.
5. A four-digit palindromic number is divisible by 9. Its first digit is 2. What is the number?
The number is 2772. A four-digit palindrome has the form ABBA. With the first digit as 2, the form is 2BB2. For divisibility by 9, the digit sum must be divisible by 9: 2 + B + B + 2 = 4 + 2B must be divisible by 9. Testing values: if 2B = 5, B = 2.5 (not a whole number). If 4 + 2B = 18, then 2B = 14, so B = 7, giving 2772. Checking: 2 + 7 + 7 + 2 = 18, which is divisible by 9. And 2772 divided by 9 = 308. So 2772 is the answer. The divisibility rule for 9 (digit sum divisible by 9) combined with the ABBA structure reduces the problem to a single equation with one unknown, which is what makes this type of question tractable under exam conditions. Knowing divisibility rules fluently is one of the most useful time-saving tools across the whole of 11+ maths.