in a cercle i have 8 numbers, a mystery number then 173, 268, 934, 252,
520, 89, and 311. What is the mystery number?

Let the mystery number be N.  Then the 8 numbers are

173, 268, 934, 252, 520, 89, 311, and N


Arrange them in a circle like this, putting N for the
missing number 

          173

      N        268

   311     o     934

     89         252

          520

We look for a pattern.

Compare the numbers that are directly opposite each
other in the circle.

173 is opposite 520
268 is opposite 89
934 is opposite 311
252 is opposite N

I first notice that 

(1)  520 is a little bit more than three times 173
     since 520/173 = 3.005780347
(2)  268 is a little bit more than three times 89  
     since 268/69 = 3.011235955
(3)  934 is a little bit more than three times 311 
     since 934/311 = 3.003215434

So I would guess either:

(A) that 252 is a little bit more than three times N

or

(B) that N is a little bit more than three times 252.

But that doesn't answer my question.

So then I ask "In each case, just how much more is the
larger than 3 times the smaller?

Let's find out:

(1)  173 is opposite 520, which is a little more than
     3 times 173. How much more? Well 3×173 is 519, and
     520 is 1 more than 519.

(2)  89 is opposite 268, which is a little more than 
     3 times 89. How much more? Well 3×89 is 267, and 
     268 is 1 more than 267.

(3)  934 is opposite 311, which is a little more than 
     3 times 311. How much more? Well 3×311 is 933, and 
     934 is 1 more than 933.

So we have the pattern.  For any pair of opposite numbers in 
the circle, the larger is 1 more than three times the smaller.  

So I want to find N, the number opposite 252.

What I need to find out now is whether N is the smaller and 252
the larger, or whether 252 is the smaller and N the larger. 

If N is the smaller then 252 should be 1 more than 3 times N.

Let's see if that's possible.

252 = 3N + 1

252 = 3N
   
  N = 251/3 = 83.66666...

No, that's not possible since we would expect N to be a whole 
number like the other seven numbers, not a decimal or fraction.

So 252 is the smaller and should be 1 more than 3 times 252.

3 times 252 is 756 and 1 more than 756 is 757

Let's see if that's possible.

252 = 3N + 1

252 = 3N
   
  N = 251/3 = 83.66666...

No, that's not possible since we would expect N to be 
a whole number like all the others, not a decimal or 
fraction.       

That only leaves the possibility that N is the larger 
and is 1 more than three times 252.  Three times 252 
is 756.  So N must be 1 more than that,
so N = 757.

Edwin
AnlytcPhil@aol.com