Reach 216000000

Logic Level 3

Dan and Sam play a game in which the first to start says the number 1, the next says 2, and the one who's next must say an integer number between the number previously said and its cube (but not including).

For example, Dan begins saying 1, then Sam says 2, and then Dan can say whichever number he wants between 2 and 8; that is, he can reply 3, 4, 5, 6 or 7.

The game finishes when someone reaches 216000000 (who is the winner ). If Dan begins, who will win? This means, who has a winning strategy?

This is the ninth problem of the set Winning Strategies .

Sam Both Dimitri Neither Dan

2 solutions

Abhay Tiwari
Apr 29, 2016

The one who will say 216000000, will win. But, who is the one that will say 216000000?.
The one who will say its cube root, $(216000000)^{\frac{1}{3}}=600$ . But 600 isn't the cube of any integer, the closest cube to it is $512$ .
So, the one saying the number $512$ wins. Again who can possibly say the number $512$ ?.
The one who says the cube root of $512$ i.e. $(512)^{\frac{1}{3}}=8$ . Again who will say the number $8$ ?.
The one who will say its cube root .i.e $(8)^{\frac{1}{3}}=2$ . So, the one who says 2 will win, and in this case. $Sam$ is the one who says the number $2$ .
So $\boxed{SAM}$ is the winner. Congratulations!!! $Sam$ :).

I would not say whoever say 512 will win, just that who force the opponent to say 9-511 will win

Tran Hieu - 5 years, 1 month ago

Tran Hieu, nice thinking and analysis. I understand what you are trying to say here, as I myself thought the same thing, but to present the solution in a more simpler way, I thought to write as I wrote. Thanks a lot! :), for bringing this point

Abhay Tiwari - 5 years, 1 month ago

Ajinkya Shivashankar
Oct 28, 2016

A person who can choose $x$ can ensure that he can get $x^3$ to $x^3+3x^2+3x$ irrespective of the choice of the other player.

Thus if a person needs to choose any number, he must make sure to choose its cuberoot rounded down!

$\sqrt[3]{216000000}=600$ , $\sqrt[3]{600}=8.4$ , $\sqrt[3]{8}=2$

So who chooses 2 will choose 2160000000

Reach 216000000

This is the ninth problem of the set Winning Strategies .

2 solutions

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