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I have a circuit X which has 4 inputs A0,B0,A1,B1 and 2 outputs A2,B2. For n of A,B bits, I need them to pass through X. I'm trying to build an equation on how many circuits X I need for n bits. For example, for n=8 we need 7 circuits:

enter image description here

I found out that for 2<=n<=7 we need n-1 X cuirits, but I know that for n=32 we need to get 30. What is the right formula?

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    \$\begingroup\$ So each level is half the number of the previous level. Now sum these up. Hint: Geometric progression. \$\endgroup\$ – Eugene Sh. Jul 16 '19 at 17:49
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    \$\begingroup\$ I don't understand what the "X" boxes are supposed to do. How is the output of an X box related to the input of that box? How do you know that you need 30 X boxes for n=32, because I am skeptical of that answer. \$\endgroup\$ – Elliot Alderson Jul 16 '19 at 17:50
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    \$\begingroup\$ Indeed, for 32 bits you need 15 boxes. At least if the other information in the question is correct. \$\endgroup\$ – Eugene Sh. Jul 16 '19 at 17:52
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Each of your X boxes reduces 4 input signals to 2.

To put it another way, each X box reduces the total number of signals by 2.

This result is the same whatever arrangement of X boxes you use: a fat tree (like the one you drew) or a long thin chain.

If you have 2n inputs, you will always need n-1 boxes if you want only 2 outputs.

Your “n=32” solution must therefore be wrong somehow.

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