An interesting property of linear sequences - inspired by the 1% club.

The 1% club is one of my favourite quiz shows. It is the only quiz show I have actually applied to be on (no success unfortunately) but I play along on the app all the time, and also regularly complete the daily question that comes through the app. Yesterday (27th January 2026) had a very interesting question (from a maths point of view) that sparked a little dive into linear sequences. I resisted posting it yesterday as I didn't want to provide spoilers for any readers that also play along.

So, the 1% club daily question yesterday was this: 

What two digit number replaces the question marks in this sequence of numbers:

92, 23, 53, 83, 14, 44, ??

What made this interesting was the way I achieved the correct answer was very different to the way the app explained how to arrive at the answer (if you want to try and answer before I reveal the solution then don't scroll down too far!)

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The correct answer was 74. The reasoning the app gave was that if you reverse the digits of the list you get the sequence 29, 32, 35, 38, 41, and 44 and so the next value would be 47 which, when reversed gives 74. Which makes perfect sense. But it isn't how I arrived at 74.

I (as I am sure many other readers also) noticed that a lot of the jumps were +30 and that those that weren't were -69. There also seemed to be a regularity to when these jumps appeared; a jump of -69 followed by two jumps of +30. Given the jump of -69 from 83 to 14, I reasoned there would be a jump of +30 (although I was wrong about the regularity of the pattern of jumps as the next would actually be another -69).

Of course, once I realised that these two approaches both gave the same answer, I absolutely had to try and decide whether this was a property of this particular set of numbers, or whether it would be true for the reverse digits of all linear sequences made of two digit numbers.

Rather than diving in with the algebra straight the way (that is coming, don't worry), I decided to play with a few more sequences first to create further examples and see if this sequence was obviously a unique case (a very good problem solving strategy in general I find to allow for pattern spotting).

So I tried 30, 34, 38, 42, 46, 50 becoming 03, 43, 83, 24, 64, 05 - which quickly disabused me that there was any regularity to when a sequence went up or down, and then I tried 17, 24, 31, 38, 45, 52 becoming 71, 42, 13, 83, 54, 25.

It was at this point that I realised that the value of the differences were always 99 apart in the reversed sequences, in the first 30 and 69, in the second 40 and 59, in the third 70 and 29. It took me an embarrassingly long time to recognise that the subtractions were happening when the original linear sequence bridged a 10, or that if the linear sequence was going up in 3 (say) that the reversed sequence should be going up in 30.

I started to explore the algebra at this point a little, but quickly realised that I was getting confounded by the fact that I had only tried differences in the original linear sequences that were less than 10, so I tried 26, 39, 52, 65, 78, 91 becoming 62, 93, 25, 56, 87, 19 (which showed me it wasn't so simple as subtractions occurring when the original sequence bridged a 10, but was more about the units digit becoming smaller - which should have been obvious really) and also 12, 35, 58, 81 becoming 21, 53, 85, 18. This confirmed that the sum to 99 was still a thing - or more precisely that the subtractions were the positive differences subtract 99.

At this point I dived properly into the algebra, which I did as follows (again, if you want to try it first then don't scroll down):

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(I added some text to show clearly what the algebra implied that I didn't write in my own scribblings).

In terms of this as a task for pupils, I think there would be something interesting in offering KS3 pupils a chance to explore 'reverse linear' sequences - probably at a distance from linear sequences themselves. I think it might reinforce some properties of linear sequences and it would be very interesting to see if they spot the 99 link and how they try and justify it.

I definitely think there would be something about using the proof with a GCSE/Further GCSE/A-Level class, either as an example of constructing a logical proof or as an exercise for them as part of their practise in creating a deductive proof.

Of course, the question remains about what happens with linear sequences that stray into 3 digit numbers (single digits are trivial as we can just treat them as two digit numbers with first digit 0). I have answered this question to my own satisfaction and so will leave it as an exercise for the interested reader with one hint, which comes from when I shared the initial problem with other maths teachers at Twinkl and one of them came up with a third approach to the original problem (which is equivalent to what I have outlined and also leads to the correct answer):

"Add 30 each time but if the answer goes over 100 add the 100s digit to the ones digit".