Project Euler Forum

Am I missing something here? Let's say n=1. Oscar writes "10". Eric can't write any number without exceeding 2. There is one 1. Oscar wins. Yet in the problem statement, Eric guarantees a win.

Same here. The number a player can choose is probably limited with n. Not sure though.

"First, they agree on a positive integer n, and they begin by writing its binary representation on a blackboard."

So for n = 1, they start by writing "1". Then it's Oscar's turn, but his only option is to write another "1", leaving an even number of 1s on the board.

Got it. That's the part I was missing. Thanks.

Are the values in the example already mod 10^9 + 7?

The value given for S(1234) is. The value for S(12) isn't (it's using = instead of $\equiv$). Although in this case S(12) is much less than 1000000007, so it doesn't make any difference.

It's common for programmers to see mod only as an operator, because that's the way the concept is expressed in programming languages. In that case we'd say S(1234) % 1000000007 = 690421393. Or, with more words, S(1234) modulo 1000000007 is 690421393.

In mathematics the it's usually done differently, and more generally. Saying $a \equiv b$ is saying that a and b are equivalent, for some kind of equivalence. The parenthetical (mod 1000000007) tells us what that equivalence is: values are considered equivalent if they differ by a multiple of 1000000007. We could just as correctly write $S(1234) \equiv 1690421400 \pmod{1000000007}$ or $S(1234) \equiv -309578614 \pmod{1000000007}$.

The problem statement is using both: equivalence for the S(1234) example, then an operator for S(12345678).

Thank you... I may be overthinking it I will come back to it... 2 days I am annoying myself with that one. but like with many things it will come to me when I am doing something else not thinking about it. and a sound will go off in my head like uhh derrrp ... I am going to write me some ASM libs for personal bit use. thanks for the clarification.