# A logic puzzle

A logic puzzle I just found online:
$$5+3+2 = 151022\\ 9+2+4 = 183652\\ 8+6+3 = 482466\\ 5+4+5 = 202541\\ 7+2+5 = ?$$
143547
The first two digits are the first two numbers multiplied.
The middle two digits are the first and last number multiplied.
The last two digits are the first two digits plus the middle two digits minus the second number.
Edit: The answer to the problem is not 14.
Please post how long did the problem take you to solve Last edited:

berkeman
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Edit: The answer is not 14.
Agreed, it did not take me 14 seconds to solve this.

• Agreed, it did not take me 14 seconds to solve this.
How long did it take you?

fresh_42
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1 is a correct answer, that takes about 0.65 s on average: 0.5 s to grasp the kind of question, 0.15 s average reaction time.
Difficult to say how long it took @berkeman to find the pun. This varies too wide across the species. E.g. I'm bad at this and it usually takes me several thoughts.

Now do you know, why 1 is a correct answer?

1 is a correct answer, that takes about 0.65 s on average: 0.5 s to grasp the kind of question, 0.15 s average reaction time.
Difficult to say how long it took @berkeman to find the pun. This varies too wide across the species. E.g. I'm bad at this and it usually takes me several thoughts.

Now do you know, why 1 is a correct answer?
Maybe the pattern can’t “hold” like the number sequence? Every answer is possible.

fresh_42
Mentor
Maybe the pattern can’t “hold” like the number sequence? Every answer is possible.
I don't understand the question, but your second sentence is correct.

a) We have a code of the type ##(p,q,r;N)## with natural numbers.
b) We also have the implicit assumption that there is a function ##N=f(p,q,r)##.
c) The question is therefore: Given ##4## points of the graph of the function, what is the function value at a fifth point?

Now how many functions are out there, which have at least these four points in common?
Even if we assume, that ##f(x_1,x_2,x_3)## is a multivariate polynomial, say ##f(x_1,x_2,x_3)= \sum_{n_1+n_2+n_3=m} a_{n_1n_2n_3}x_1^{n_1}x_2^{n_2}x_3^{n_3}## then how many functions of this type satisfy finitely many given conditions ##f(p,q,r)=N##, in our case five?

Have a look at ##f(x,y)=5x^3+2xy^2-10x^2y^3+3x-4y##:
https://www.wolframalpha.com/input/?i=f(x,y)=5x^3+2xy^2-10x^2y^3+3x-4y

Now imagine such a graph with some points fixed. How many such surfaces are still possible which contain those points? We even have an additional dimension, an arbitrary high degree for our multivariate polynomial, and this special form for ##f(x_1,x_2,x_3)## has already been a deliberate restriction. Imagine if we add all trigonometric functions, exponential function, roots of any degree or whatever.

PeroK
Homework Helper
Gold Member
2020 Award
1 is a correct answer, that takes about 0.65 s on average: 0.5 s to grasp the kind of question, 0.15 s average reaction time.
Difficult to say how long it took @berkeman to find the pun. This varies too wide across the species. E.g. I'm bad at this and it usually takes me several thoughts.

Now do you know, why 1 is a correct answer?

This misses the point. This could be a code. In which case, breaking the code is to find a simple pattern to the numbers. Last year someone showed me a copy of the GCHQ puzzle book.

https://www.waterstones.com/book/the-gchq-puzzle-book/gchq/9780718185541

This is the sort of puzzle you'll find in there. These puzzles do not generally involve pure mathematical proofs, but a mixture of maths, logic, abduction and general knowledge.

fresh_42
Mentor
This misses the point.
No. It criticizes the way it is posed##{}^*)##: Find the answer! It should read: Find all hidden assumptions! Or if you like and suggested: Break the code!
To claim there is a unique answer might be o.k. on facebook, within a mathematical context it is not.

##{}^*)## This refers to the general way those questions are asked, not the specific one by the OP.

Edit: Alternatively the question would have to be a minimum problem: Find the shortest (including the reason) answer, in which case the measure must be named and the minimality must be proven. Or the requirement ##f\, : \,\mathbb{N}^3 \longrightarrow \mathbb{N}## should be made. But then uniqueness becomes a big problem without further conditions (I guess).

Last edited:
Staff Emeritus
If you restrict f to be of no higher degree than bilinear in p,q, and r is it true that N can be any real number?

fresh_42
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If you restrict f to be of no higher degree than bilinear in p,q, and r is it true that N can be any real number?
The supposed solution isn't linear in any variable, only affine linear. If we restrict the allowed powers of ##p,q,r## to be at most ##1##, then it depends on what ##p,q,r## are allowed to be. E.g. the answer is yes, if ##r \in \mathbb{R},## although it would not necessarily contain the four points. If we restrict ##f## on integers both on domain and codomain, then we quickly get a question like FLT, and that took about ##350## years and a very persistent genius to solve.

fresh_42
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Since we are talking more about the properties of such questions than giving the intended answers, I will close the thread.
It is a fun question, but mathematically underdetermined.

• berkeman