Delphi Flip

A and B play a game. A has originally $1$ gram of gold and B has an unlimited amount.
Each round goes as follows:

A chooses and displays, $x$, a nonnegative real number no larger than the amount of gold that A has.

Either B chooses to TAKE. Then A gives B $x$ grams of gold.

Or B chooses to GIVE. Then B gives A $x$ grams of gold.

B TAKEs $n$ times and GIVEs $n$ times after which the game finishes.

Define $g(X)$ to be the smallest value of $n$ so that A can guarantee to have at least $X$ grams of gold at the end of the game. You are given $g(1.7) = 10$.

Find $g(1.9999)$.

This problem might be challenging because it calls for a strategic perspective and understanding the trade-off between risk and return.

The goal of player A is to gain more gold without losing too much. Player B will aim to minimize A’s gain.

Because gold can be divided indefinitely, A can effectively choose the stake for each round. It’s important to consider here that in every round, A has full discretion over how much gold will be at stake, regardless of B’s decision to TAKE or GIVE.

To find the smallest value of n such that A can guarantee to have at least X grams of gold by the end of the game, a strategic approach is required. Ideally, A should start by putting up a small stake to limit the loss in the event that B chooses to TAKE. With each round, A can increase the stake gradually to accumulate more gold when B chooses to GIVE.

In the given game scenario, g(1.7) = 10. This means that after ten rounds of A choosing the right stake and getting the right responses from B, A can secure at least 1.7 grams of gold. Note here that this doesn’t necessarily mean that A attains exactly 1.7 grams after the tenth round. g(1.7) simply means that the tenth round is the earliest that A can ensure a minimum of 1.7 grams.

To find g(1.9999), we need to find the smallest number of rounds in which A can guarantee having at least 1.9999 grams of gold. Here A needs to be more conservative in raising the stake, given that B has unlimited gold to give and A’s gold is initially only 1 gram.

If, for example, A starts with 0.1 grams per round and B gives in each of these rounds, A will have 2 grams of gold after the tenth round. But if A increases the stake to 0.11 grams per round, B can take in the first round and then give in the next nine rounds, leaving A with less than 2 grams of gold by the tenth round. So, to guarantee having at least 1.9999 grams, A must carefully choose the stake to ensure that even if B takes the gold in some rounds, A will still reach the target.

The actual solution for g(1.9999) needs mathematical optimization, which is beyond the simple calculation and would require programming or advanced mathematical software.

To sum up, the value of g(X) is determined by how A strategically chooses his/her stake per round so as to ensure a guaranteed gain of X grams in the minimal number of rounds, considering that B can either take or give gold. For larger values of X, A has to be more conservative and strategic so that the number of rounds is minimized.

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