Here's a simple strategy for length 4. You didn't say who goes first; I"ll prove the stronger version where B goes first. Call the first play by B b1, the second b2, whiile A's first play is 0 (if b1= 0, just translate everything). Choose X>1 coprime to b1 and b2. All subsequent plays by A will be on multiples of X, so we can ignore all plays that are not multiples of X, including B1 and B2. A's second play is at 2X. If B3 is anything other than -4X, -2X, 4X, or 6X, A can win by playing A3 at either -2X or 4X, with two threats of 4 in a row. If B plays at one of these, A plays at X, and at -X or 3X next. A proof (from a known theorem) that A wins for length 5. It is well-known by those who know about such things that the first player has a forced win at the original version of the game of Go-Moku, played on a 2-d array, where 5 in a row (orthogonally or diagonally) wins, on a sufficiently large board. use the above trick to make B's first move ignorable, then number a sufficiently large square board in the obvious way. This game is easier for A to win than simplified Go-Moku in many ways: 5 in a row for B does not win for B, and many sets of 5 that are not 5-in-a-row by the Go-Moku definition are still a win (5 in a row along a direction that is not orthogonal or diagonal, such as a knight's move, or every other spot along a row or a column). But every 5-in-a-row by Go-Moku rules is also an AP, so when A makes 5 in a row, he wins both games. I suspect A can force a win with lengths much greater than 5. In particular, if A can force k in a row on an N-dimensional board, that suffices to get an AP of length k, and the higher the dimension, the easier it is to force long rows. I think Winning Ways says something about n-dimensional k-in-a-row, but I don't have my copy handy. If I had to guess, I would guess that A can force an arithmetic progression of any fixed finite length. Andy On Thu, Aug 13, 2015 at 11:58 AM, Allan Wechsler <acwacw@gmail.com> wrote:
I'm pretty sure I have a strategy for length four now. The length three row made by the previous strategy is only blocked by B on one end; A can extend it to four on the other.
On Thu, Aug 13, 2015 at 11:43 AM, Allan Wechsler <acwacw@gmail.com> wrote:
B cannot prevent A from owning two consecutive integers on the second move. Regardless of what B does, A can now place a "pivot" far enough away from the first pair that the reflection of the pair through the pivot is a distant pair of integers, neither of which are claimed. B cannot claim both, and A wins on the fourth move.
I haven't thought much about length four goals. This is very reminiscent of the "angels and demons" problem.
On Thu, Aug 13, 2015 at 8:59 AM, David Wilson <davidwwilson@comcast.net> wrote:
You'll have to forgive me. Having encountered the abbreviation "A.P." in high school, I assumed it was on a par with "GCD" or "LCM" in terms of general familiarity, meaning I didn't need to define it in this forum. Apologies. "A.P." indeed stands for "arithmetic progression".
I feel your pain. When I see something I don't understand in this forum, I usually wait for the follow-up messages to clarify the situation, or in the case of analysis geeks spouting about q-equations, Galois groups and hypergeometric functions, politely skip on to the next email. "Better to remain silent and be thought a fool than to speak out and remove all doubt."
But seriously, in any mathematical discussion you must assume some background material, which implies that someone is not going to understand what you're talking about.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Dan Asimov Sent: Thursday, August 13, 2015 1:01 AM To: math-fun Subject: Re: [math-fun] Game question
David,
I can't remember the last time I saw A.P. used as an abbrev. to mean arithmetic progression — quite possibly I never did. It took me maybe 3 minutes before I thought of that (which I presume is what you mean, right?).
Maybe next time you can spell it out — or abbreviate it like "arith. prog." — the first time with e.g. "A.P." in parentheses, and then use A.P. after that?
(At first I thought it was probably some common abbrev. in number theory, a subject I don't have a lot of knowledge about.)
Thanks,
Dan
On Aug 12, 2015, at 9:22 PM, David Wilson <davidwwilson@comcast.net> wrote:
Consider a game in which two players, A and B, each choose distinct integers by turn.
A's object is to maximize the length of the longest A.P. among his selected integers.
B's object is to limit the length of A's longest A.P.
Show that B cannot prevent A from obtaining an A.P. of length 3.
Can B prevent A from obtaining an A.P. of some length N?
What is the smallest such N?
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