31

The answer is because


12



8

The numbers are: Step 1: Step 2 Step 3 Step 4 Step 5


7

We know that: It then follows that So the question boils down to finding The solutions are therefore:


6

Horst can place That's because Note


6

Alice chooses columns, Bob chooses rows in this matrix: $$ \pmatrix{ * & r & p & s & a \\ r&0&1&-1&0\\ p&-1&0&1&1\\ s&1&-1&0&-1\\ a&0&-1&1&0 } $$ $+1$ means Alice wins, $0$ is tie, $-1$ Bob wins. There is no Nash equilibrium. According to this solver/website https://cgi.csc.liv....


5

The answer to the puzzle is: Puzzle 1: Puzzle 2: Puzzle 3: Putting it all together:


4

Answer: Proof:


4

The answers for $M$ are somewhat surprising! I expected smaller numbers. $k=0$ Minimum Maximum $k=1$ Minimum Maximum $k=2$ Minimum Maximum


4

Perhaps the answer is Because And the fact that We can deduce the answer: Which matches Hint 6 in that (Those not after "-" is the original text):


3

The answer maybe: The reason is:


2

Assuming that every digit left of the '=' operatore is a symbolic representation of a value, then we would have: '4' -> 4 '2' -> 2 '6' -> 6 '8' -> 24 '5' -> 85 '7' -> 221 then '8'+'6' (symbolically) would become 24+6 (numerically), therefore '8'+'6' = 30 If this is the key, then there would be several alternative solutions, because we could assign ...


2

So we can break up the solution as which decodes as Although I suspect there may be another level of decoding and the final answer should be


1

Partial Answer 1. 2. 3. So overall I think


1

Not an answer, but perhaps a point in the right direction.


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