How can S(3) ever be 35?

The total number of function to select from (B^3 -> B) is 16 and 35 is greater than that !

Here goes a full table of all function B^3 -> B

TTT T

TTT F

TTF T

TTF F

TFT T

TFT F

TFF T

TFF F

FTT T

FTT F

FTF T

FTF F

FFT T

FFT F

FFF T

FFF F

Groetjes Albert

## Problem 703

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Comments, questions and clarifications about PE problems.

### Re: Problem 703

Each function from B^3 to B maps three inputs to one output. There are 8 possibilities for the input, and a function is specified by giving an output independently for each of them. There are 2^8 = 256 ways of doing this, so there are 256 possible functions. In general, there are 2^(2^n) functions from B^n to B.

For two inputs, there are 16 functions. Here they are:

For two inputs, there are 16 functions. Here they are:

Code: Select all

```
A B out A B out A B out A B out
0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 1 0 0 1 0 0 1 0
1 0 0 1 0 0 1 0 1 1 0 1
1 1 0 1 1 1 1 1 0 1 1 1
A B out A B out A B out A B out
0 0 0 0 0 0 0 0 0 0 0 0
0 1 1 0 1 1 0 1 1 0 1 1
1 0 0 1 0 0 1 0 1 1 0 1
1 1 0 1 1 1 1 1 0 1 1 1
A B out A B out A B out A B out
0 0 1 0 0 1 0 0 1 0 0 1
0 1 0 0 1 0 0 1 0 0 1 0
1 0 0 1 0 0 1 0 1 1 0 1
1 1 0 1 1 1 1 1 0 1 1 1
A B out A B out A B out A B out
0 0 1 0 0 1 0 0 1 0 0 1
0 1 1 0 1 1 0 1 1 0 1 1
1 0 0 1 0 0 1 0 1 1 0 1
1 1 0 1 1 1 1 1 0 1 1 1
```

- kenbrooker
**Posts:**140**Joined:**Mon Feb 19, 2018 3:05 am**Location:**Northern California, USA

### Re: Problem 703

For those more conversant in Computer Science than Mathematics, the following translation of Problem 703 may be helpful, if not invaluable -- A k-input binary Truth Table is a map from k input bits (0=False; 1=True) to 1 output bit. For example, the 2-input binary Truth Tables for the logical AND and XOR functions are: x y xANDy xXORy - - ------- ------- 0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 0 As a further example, we can demonstrate that three 2-input binary Truth Tables, T, satisfy the formula T(x, y) AND T(y, xANDy) = 0 for all 2-bit inputs (x, y) as follows: Tables(x, y) x, y [A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P] 0, 0 [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1] 0, 1 [0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1] 1, 0 [0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1] 1, 1 [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1] z = xANDy Tables(y, z) y, z [A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P] 0, 0 [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1] 1, 0 [0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1] 0, 0 [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1] 1, 1 [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1] &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& T(x, y) AND T(y, z) x, y,(z) [A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P] 0, 0, 0 [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1] 0, 1, 0 [0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1] 1, 0, 0 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1] 1, 1, 1 [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1] Tables A, C & E all zero for a Count = 3 Additionally, there are 2118 4-input binary Truth Tables, T, that satisfy the formula T(a,b,c,d) AND T( (b,c,d,a) AND (bXORc) ) = 0 for all 4-bit inputs (a,b,c,d). How many 20-input binary truth tables, T, satisfy the formula T(a,b,c, ... t) AND T( (b,c,d, ... t,a) AND (bXORc) ) = 0 for all 20-bit inputs (a,b,c, ... t)?

"

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