Prove by induction that if $n equiv 0 pmod{4}$, then there exists a strategy that one player can win the...
Is this a real picture of Jordan Peterson in New Zealand with a fan wearing a shirt that says "I'm a Proud Islamaphobe"?
Identifying the interval from A♭ to D♯
Can a druid choose the size of its wild shape beast?
Dice rolling probability game
Do I need life insurance if I can cover my own funeral costs?
Is it true that good novels will automatically sell themselves on Amazon (and so on) and there is no need for one to waste time promoting?
Min function accepting varying number of arguments in C++17
Brexit - No Deal Rejection
Is it possible to upcast ritual spells?
How to read the value of this capacitor?
If curse and magic is two sides of the same coin, why the former is forbidden?
How could a scammer know the apps on my phone / iTunes account?
Time travel from stationary position?
What did Alexander Pope mean by "Expletives their feeble Aid do join"?
Happy pi day, everyone!
What are substitutions for coconut in curry?
How to deal with a cynical class?
A sequence that has integer values for prime indexes only:
Bach's Toccata and Fugue in D minor breaks the "no parallel octaves" rule?
How to create the Curved texte?
Why one should not leave fingerprints on bulbs and plugs?
What options are left, if Britain cannot decide?
Who is flying the vertibirds?
Why is the President allowed to veto a cancellation of emergency powers?
Prove by induction that if $n equiv 0 pmod{4}$, then there exists a strategy that one player can win the game
Variation of Nim: Player who takes last match losesProve using a strategy stealing argument that player 1 has a winning strategy in the chomp gameInduction solution for game of coinsNim Variant - Strong Induction ProofHow do I prove using strong form induction a statement regarding winning strategies in this coin game?Misere nim, 2nd player winning strategy proof by inductionWho should win the game dependent on $x$ and $y$?Consider the following two player game. A pile of coins is places on a tableStrong induction: Game of NimProof by induction of a 2 player dot removal game
$begingroup$
Alice and Bob play a game starting with a pile of $nge{1}$ sticks. Each player on his or her turn can remove $1$, $2$ or $3$ sticks from the pile. The last player to stick wins. Alice first. How do I prove by induction that if $n equiv 0 pmod{4}$, then there exists a strategy that Bob can follow which guarantees he will win no matter what moves Alice makes?
Not sure how to do this, any help is appreciated!
modular-arithmetic induction
edited Mar 10 at 23:24
rtybase
11.4k31533
asked Mar 10 at 22:35
macymacy
436
$endgroup$
add a comment |
$begingroup$
Alice and Bob play a game starting with a pile of $nge{1}$ sticks. Each player on his or her turn can remove $1$, $2$ or $3$ sticks from the pile. The last player to stick wins. Alice first. How do I prove by induction that if $n equiv 0 pmod{4}$, then there exists a strategy that Bob can follow which guarantees he will win no matter what moves Alice makes?
Not sure how to do this, any help is appreciated!
modular-arithmetic induction
edited Mar 10 at 23:24
rtybase
11.4k31533
asked Mar 10 at 22:35
macymacy
436
$endgroup$
$begingroup$
Try it for some small case and see if you can come up with a strategy for Bob to win. I.e, try $n = 16$ for example, and work backwards from $1$ stick and see if you can make it so Bob wins.
$endgroup$
– rubikscube09
Mar 10 at 22:38
add a comment |
$begingroup$
Alice and Bob play a game starting with a pile of $nge{1}$ sticks. Each player on his or her turn can remove $1$, $2$ or $3$ sticks from the pile. The last player to stick wins. Alice first. How do I prove by induction that if $n equiv 0 pmod{4}$, then there exists a strategy that Bob can follow which guarantees he will win no matter what moves Alice makes?
Not sure how to do this, any help is appreciated!
modular-arithmetic induction
edited Mar 10 at 23:24
rtybase
11.4k31533
asked Mar 10 at 22:35
macymacy
436
$endgroup$
Alice and Bob play a game starting with a pile of $nge{1}$ sticks. Each player on his or her turn can remove $1$, $2$ or $3$ sticks from the pile. The last player to stick wins. Alice first. How do I prove by induction that if $n equiv 0 pmod{4}$, then there exists a strategy that Bob can follow which guarantees he will win no matter what moves Alice makes?
Not sure how to do this, any help is appreciated!
modular-arithmetic induction
modular-arithmetic induction
edited Mar 10 at 23:24
rtybase
11.4k31533
asked Mar 10 at 22:35
macymacy
436
edited Mar 10 at 23:24
rtybase
11.4k31533
asked Mar 10 at 22:35
macymacy
436
edited Mar 10 at 23:24
rtybase
11.4k31533
edited Mar 10 at 23:24
rtybase
11.4k31533
edited Mar 10 at 23:24
rtybase
11.4k31533
11.4k31533
asked Mar 10 at 22:35
macymacy
436
asked Mar 10 at 22:35
macymacy
436
asked Mar 10 at 22:35
macymacy
436
436
$begingroup$
Try it for some small case and see if you can come up with a strategy for Bob to win. I.e, try $n = 16$ for example, and work backwards from $1$ stick and see if you can make it so Bob wins.
$endgroup$
– rubikscube09
Mar 10 at 22:38
add a comment |
$begingroup$
Try it for some small case and see if you can come up with a strategy for Bob to win. I.e, try $n = 16$ for example, and work backwards from $1$ stick and see if you can make it so Bob wins.
$endgroup$
– rubikscube09
Mar 10 at 22:38
$begingroup$
Try it for some small case and see if you can come up with a strategy for Bob to win. I.e, try $n = 16$ for example, and work backwards from $1$ stick and see if you can make it so Bob wins.
$endgroup$
– rubikscube09
Mar 10 at 22:38
$begingroup$
Try it for some small case and see if you can come up with a strategy for Bob to win. I.e, try $n = 16$ for example, and work backwards from $1$ stick and see if you can make it so Bob wins.
$endgroup$
– rubikscube09
Mar 10 at 22:38
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
For $;n=4;$ , no matter how many sticks Alice removes, in his turn Bob wins (make this more formal if necessary). Assume this is true for any amount$;=0pmod 4;$ that is less than $;n;$, with $;n=0pmod4;$ . So Alice begins, and in hist first move Bob remove as many sticks as necessary to leave $;n-4;$ on the game...and voila! By the inductive hypothesis and since it is Alice's turn, Bob will always be able to win the game.
answered Mar 10 at 22:39
DonAntonioDonAntonio
179k1494233
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3143009%2fprove-by-induction-that-if-n-equiv-0-pmod4-then-there-exists-a-strategy-t%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
For $;n=4;$ , no matter how many sticks Alice removes, in his turn Bob wins (make this more formal if necessary). Assume this is true for any amount$;=0pmod 4;$ that is less than $;n;$, with $;n=0pmod4;$ . So Alice begins, and in hist first move Bob remove as many sticks as necessary to leave $;n-4;$ on the game...and voila! By the inductive hypothesis and since it is Alice's turn, Bob will always be able to win the game.
answered Mar 10 at 22:39
DonAntonioDonAntonio
179k1494233
$endgroup$
add a comment |
$begingroup$
For $;n=4;$ , no matter how many sticks Alice removes, in his turn Bob wins (make this more formal if necessary). Assume this is true for any amount$;=0pmod 4;$ that is less than $;n;$, with $;n=0pmod4;$ . So Alice begins, and in hist first move Bob remove as many sticks as necessary to leave $;n-4;$ on the game...and voila! By the inductive hypothesis and since it is Alice's turn, Bob will always be able to win the game.
answered Mar 10 at 22:39
DonAntonioDonAntonio
179k1494233
$endgroup$
add a comment |
$begingroup$
For $;n=4;$ , no matter how many sticks Alice removes, in his turn Bob wins (make this more formal if necessary). Assume this is true for any amount$;=0pmod 4;$ that is less than $;n;$, with $;n=0pmod4;$ . So Alice begins, and in hist first move Bob remove as many sticks as necessary to leave $;n-4;$ on the game...and voila! By the inductive hypothesis and since it is Alice's turn, Bob will always be able to win the game.
answered Mar 10 at 22:39
DonAntonioDonAntonio
179k1494233
$endgroup$
For $;n=4;$ , no matter how many sticks Alice removes, in his turn Bob wins (make this more formal if necessary). Assume this is true for any amount$;=0pmod 4;$ that is less than $;n;$, with $;n=0pmod4;$ . So Alice begins, and in hist first move Bob remove as many sticks as necessary to leave $;n-4;$ on the game...and voila! By the inductive hypothesis and since it is Alice's turn, Bob will always be able to win the game.
answered Mar 10 at 22:39
DonAntonioDonAntonio
179k1494233
answered Mar 10 at 22:39
DonAntonioDonAntonio
179k1494233
answered Mar 10 at 22:39
DonAntonioDonAntonio
179k1494233
answered Mar 10 at 22:39
DonAntonioDonAntonio
179k1494233
179k1494233
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3143009%2fprove-by-induction-that-if-n-equiv-0-pmod4-then-there-exists-a-strategy-t%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
Try it for some small case and see if you can come up with a strategy for Bob to win. I.e, try $n = 16$ for example, and work backwards from $1$ stick and see if you can make it so Bob wins.
$endgroup$
– rubikscube09
Mar 10 at 22:38