Well done Steve, and I am confident that Ravvie has the same solution.
My picture below might help one or two browsers visualise the solution as well.
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There is a pattern to the sequence of “equations” displayed below.
What number should replace the question mark ?
1 + 4 = 5
2 + 5 = 12
3 + 6 = 21
…
8 + 11 = ?
I think one answer can legitimately be 96. Whether that’s your answer is another matter…
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That’s the problem with Brain “Teasers”, which this one is, as opposed to dressed-up GCSE maths questions, which this one isn’t 
But you do have a perfectly good, logical answer, Steve. Well done !
It also happens to be the same as mine !
No doubt there are some equally valid alternative answers, but I haven’t looked for, nor seen any.
Yes. Initially I thought four simultaneous equations with four unknowns, cunningly presented in two dimensions, surely that wasn’t in the GCSE curriculum! But actually it helps, as we can substitute as we go along, solving for A:
First row:
A + (8-A) = 8
Second row, using each column:
(13-A) - A = 6
Hence A = 3.5 and the rest follow.
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I just showed the puzzle to Mrs R. She guessed 3 then 4 for the bottom right figure and realised there was something “odd”, but they looked close. After I said that they don’t have to be whole numbers she tried 3.5 and worked the rest out.
All done in her head of course. Amazing!
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I noticed that each total is the sum of the two numbers to its left PLUS the previous total, ie a running cumulative. However it is also the result of the product of the two numbers plus the first number (ie numberA + (numberA x numberB). These two methods must be related in some way, but I haven’t worked out how yet.
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I note that Steve has nailed it already and is musing on the underlying pattern.
I went down a different path, finding something that worked for the first three equations (as genuine equations) but it broke down before the 8th row:
1 + 4 = 5
2 + 5 = 12
3 + 6 = 21
4 + 7 = 102
5 + 8 = 1101
6 + 9 = 00000 00000 00000
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My solution was based on your second method ie product of the two numbers plus the first one.
I agree there must be a link with your first option. Interesting !
I’ve sussed it now.
Don’s method for two terms n, and (n+3) creates a total which can be simplified to n(n+4).
My running cumulative method creates two arithmetic progressions, each of which can be totalled using the sum of arithmetic progression formula. This also eventually simplifies to n(n+4).
Sorry I’m not able to post my workings more clearly.
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Steve, that’s a wonderful piece of work. Thank you.
The Road from X to Y.
Our Local Council is building a new road from Village X to Village Y but hasn’t made much progress to date. In fact, they have only completed a few sections as shown on the plan below.
The numbers along the top of the plan and down the side of the plan show how many sections of road will be built in each column and each row. The planned road will not cross itself at any point.
Each cell on the Plan can only contain :-
- a straight length of road either horizontally or vertically on the plan or
- a turn to the right or a turn to the left (or up/down)
Can you complete the Plan ?
(I hope the instructions are clear, if not, just ask and I’ll try to clarify)
This is the sort of puzzle that Mrs R likes. She had 5 minutes spare last night (literally), so she had a go. I can post the pic later if you like.
All I can say is the council should hire some new civil engineers!
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Hi Ravvie,
Well done to Mrs R.
Look forward later to the picture.
Good one. These puzzles are printed every day in The Times newspaper (UK) as “train tracks”, but I haven’t attempted one in a while. Enjoyable, thanks!
Thank you Bobby. I’m pleased the puzzle regenerated your past interest. The one I posted, and a couple of others, had laid dormant for a couple of years. Looks like we shall all be buying The Times tomorrow morning
Nicely done Ravvie, or rather …… Mrs R.
Likewise Bobby, well done.
Just watching QI !
I toss two fair coins. What is the probability of them both landing heads up ?