Number machine

[jaygee]

input	output
2	6
4	7
8	9
12	11

hi. i'm hoping for a bit of a point in the right direction. there's a number machine and i've got to find out what connects the input and output. its just not that obvious (to me, anyway)

input output
2 6
4 7
8 9
12 11


i haven't had any trouble with the other number machines just this one. i hope someone can help!

 

[stapel]

input	output
2	6
4	7
8	9
12	11

hi. i'm hoping for a bit of a point in the right direction. there's a number machine and i've got to find out what connects the input and output. its just not that obvious (to me, anyway)

Hint: How far apart are the inputs? (The differences are not all the same!) How far apart are the outputs? (Again, not the same!) Can you see a relationship between these differences?

 

[HallsofIvy]

Of course, there exist an infinite number of functions that will assign 4 given numbers to 4 other numbers.

One way to do this is to note that, with 4 "conditions" you can determine 4 numbers and a general cubic, \(\displaystyle ax^3+ bx^2+cx+ d\), has four coefficients. If x= 2 then 8a+ 4b+ 2c+ d= 6. If x= 4 then 64a+ 16b+ 4c+ d= 7. If x= 8, then 512a+ 64b+ 8c+ d= 9. If x= 12, then 1728a+ 144b+ 12c+ d= 11.

That gives you four linear equations to solve for a, b, c, and d.

 

[jaygee]

Thank you - i think i've got it - half the number and add 5. Thank you so much. it was really bugging me.