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<h2 class="hd hd-2 unit-title">Homework</h2>
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<p>The exercises below will count towards your grade. <strong>You have only one chance to answer these questions. </strong>Take your time, and think carefully before answering.</p>
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<h3 class="hd hd-3 problem-header" id="9d42fc504fcf4686a2ce6324a774a066-problem-title" aria-describedby="block-v1:MITx+24.118x+2T2020+type@problem+block@9d42fc504fcf4686a2ce6324a774a066-problem-progress" tabindex="-1">
Problem 1
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<p>Imagine an island on which everyone is either a knight, who always tells the truth, or a knave, who always lies. Ten islanders, \(S_0, S_1, \ldots, S_{9}\) are lined up. \(S_0\) is in the back of the line; in front of her is \(S_1\), in front of him is \(S_2\), and so on. Every islander says: </p>
<p>"There is at least one person in front of me, and everyone in front of me is a knave." </p>
<p>Is \(S_0\) a knight or a knave?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_9d42fc504fcf4686a2ce6324a774a066_2_1">
<fieldset aria-describedby="status_9d42fc504fcf4686a2ce6324a774a066_2_1">
<div class="field">
<input type="radio" name="input_9d42fc504fcf4686a2ce6324a774a066_2_1" id="input_9d42fc504fcf4686a2ce6324a774a066_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="9d42fc504fcf4686a2ce6324a774a066_2_1-choice_0-label" for="input_9d42fc504fcf4686a2ce6324a774a066_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_9d42fc504fcf4686a2ce6324a774a066_2_1"> \(S_0\) is a knight.
</label>
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<div class="field">
<input type="radio" name="input_9d42fc504fcf4686a2ce6324a774a066_2_1" id="input_9d42fc504fcf4686a2ce6324a774a066_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="9d42fc504fcf4686a2ce6324a774a066_2_1-choice_1-label" for="input_9d42fc504fcf4686a2ce6324a774a066_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_9d42fc504fcf4686a2ce6324a774a066_2_1"> \(S_0\) is a knave.
</label>
</div>
<div class="field">
<input type="radio" name="input_9d42fc504fcf4686a2ce6324a774a066_2_1" id="input_9d42fc504fcf4686a2ce6324a774a066_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="9d42fc504fcf4686a2ce6324a774a066_2_1-choice_2-label" for="input_9d42fc504fcf4686a2ce6324a774a066_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_9d42fc504fcf4686a2ce6324a774a066_2_1"> There is no coherent assignment of knight or knave status to \(S_0\).
</label>
</div>
<div class="field">
<input type="radio" name="input_9d42fc504fcf4686a2ce6324a774a066_2_1" id="input_9d42fc504fcf4686a2ce6324a774a066_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="9d42fc504fcf4686a2ce6324a774a066_2_1-choice_3-label" for="input_9d42fc504fcf4686a2ce6324a774a066_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_9d42fc504fcf4686a2ce6324a774a066_2_1"> It is consistent with the problem that \(S_0\) be a knight, and consistent with the problem that \(S_0\) be a knave.
</label>
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<span id="answer_9d42fc504fcf4686a2ce6324a774a066_2_1"/>
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<span class="status unanswered" id="status_9d42fc504fcf4686a2ce6324a774a066_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<span id="solution_9d42fc504fcf4686a2ce6324a774a066_solution_1"/>
</div><p>Next, imagine that there are a countable infinity of islanders \(S_0, S_1, S_2, \ldots\) lined up (it's a big island &#8212; an infinitely big one). \(S_0\) is at the back of the line; in front of her is \(S_1\), in front of him is \(S_2\), and so on. Every islander says: </p>
<p>"There is at least one person in front of me, and everyone in front of me is a knave."</p>
<p>Is \(S_0\) a knight or a knave?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_9d42fc504fcf4686a2ce6324a774a066_3_1">
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<div class="field">
<input type="radio" name="input_9d42fc504fcf4686a2ce6324a774a066_3_1" id="input_9d42fc504fcf4686a2ce6324a774a066_3_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="9d42fc504fcf4686a2ce6324a774a066_3_1-choice_0-label" for="input_9d42fc504fcf4686a2ce6324a774a066_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_9d42fc504fcf4686a2ce6324a774a066_3_1"> \(S_0\) is a knight.
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<input type="radio" name="input_9d42fc504fcf4686a2ce6324a774a066_3_1" id="input_9d42fc504fcf4686a2ce6324a774a066_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="9d42fc504fcf4686a2ce6324a774a066_3_1-choice_1-label" for="input_9d42fc504fcf4686a2ce6324a774a066_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_9d42fc504fcf4686a2ce6324a774a066_3_1"> \(S_0\) is a knave.
</label>
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<div class="field">
<input type="radio" name="input_9d42fc504fcf4686a2ce6324a774a066_3_1" id="input_9d42fc504fcf4686a2ce6324a774a066_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="9d42fc504fcf4686a2ce6324a774a066_3_1-choice_2-label" for="input_9d42fc504fcf4686a2ce6324a774a066_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_9d42fc504fcf4686a2ce6324a774a066_3_1"> There is no coherent assignment of knight or knave status to \(S_0\).
</label>
</div>
<div class="field">
<input type="radio" name="input_9d42fc504fcf4686a2ce6324a774a066_3_1" id="input_9d42fc504fcf4686a2ce6324a774a066_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="9d42fc504fcf4686a2ce6324a774a066_3_1-choice_3-label" for="input_9d42fc504fcf4686a2ce6324a774a066_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_9d42fc504fcf4686a2ce6324a774a066_3_1"> It is consistent with the problem that \(S_0\) be a knight, and consistent with the problem that \(S_0\) be a knave.
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<span id="answer_9d42fc504fcf4686a2ce6324a774a066_3_1"/>
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<span id="solution_9d42fc504fcf4686a2ce6324a774a066_solution_2"/>
</div><p>Now, imagine the infinitely many islanders are lined up just as before. But this time every islander says: </p>
<p>"There is at least one person in front of me, and everyone in front of me is a knight."</p>
<p>Is \(S_0\) a knight or a knave?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_9d42fc504fcf4686a2ce6324a774a066_4_1">
<fieldset aria-describedby="status_9d42fc504fcf4686a2ce6324a774a066_4_1">
<div class="field">
<input type="radio" name="input_9d42fc504fcf4686a2ce6324a774a066_4_1" id="input_9d42fc504fcf4686a2ce6324a774a066_4_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="9d42fc504fcf4686a2ce6324a774a066_4_1-choice_0-label" for="input_9d42fc504fcf4686a2ce6324a774a066_4_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_9d42fc504fcf4686a2ce6324a774a066_4_1"> \(S_0\) is a knight.
</label>
</div>
<div class="field">
<input type="radio" name="input_9d42fc504fcf4686a2ce6324a774a066_4_1" id="input_9d42fc504fcf4686a2ce6324a774a066_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="9d42fc504fcf4686a2ce6324a774a066_4_1-choice_1-label" for="input_9d42fc504fcf4686a2ce6324a774a066_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_9d42fc504fcf4686a2ce6324a774a066_4_1"> \(S_0\) is a knave.
</label>
</div>
<div class="field">
<input type="radio" name="input_9d42fc504fcf4686a2ce6324a774a066_4_1" id="input_9d42fc504fcf4686a2ce6324a774a066_4_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="9d42fc504fcf4686a2ce6324a774a066_4_1-choice_2-label" for="input_9d42fc504fcf4686a2ce6324a774a066_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_9d42fc504fcf4686a2ce6324a774a066_4_1"> There is no coherent assignment of knight or knave status to \(S_0\).
</label>
</div>
<div class="field">
<input type="radio" name="input_9d42fc504fcf4686a2ce6324a774a066_4_1" id="input_9d42fc504fcf4686a2ce6324a774a066_4_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="9d42fc504fcf4686a2ce6324a774a066_4_1-choice_3-label" for="input_9d42fc504fcf4686a2ce6324a774a066_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_9d42fc504fcf4686a2ce6324a774a066_4_1"> It is consistent with the problem that \(S_0\) be a knight, and consistent with the problem that \(S_0\) be a knave.
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<span id="answer_9d42fc504fcf4686a2ce6324a774a066_4_1"/>
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<h3 class="hd hd-3 problem-header" id="b1da085dfd1e46e9ace7ebf7b118df43-problem-title" aria-describedby="block-v1:MITx+24.118x+2T2020+type@problem+block@b1da085dfd1e46e9ace7ebf7b118df43-problem-progress" tabindex="-1">
Problem 2
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<p>Lazy wants to run from \(A\) to \(B\), but he likes to take one-second breaks. He first stops halfway between \(A\) and \(B\) and takes a one-second break. He then stops halfway between <i>that</i> point and \(B\) and takes a one-second break, and so on. More generally, for each \(k \geq 1\), Lazy takes a break at a distance of \( (B - A)/2^k \) from \(B\). Assume that the traveling itself takes Lazy no time at all.</p>
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<legend id="b1da085dfd1e46e9ace7ebf7b118df43_2_1-legend" class="response-fieldset-legend field-group-hd">Is there a positive integer \(n\) such that after \(n\) seconds Lazy has reached point \(B\)?</legend>
<div class="field">
<input type="radio" name="input_b1da085dfd1e46e9ace7ebf7b118df43_2_1" id="input_b1da085dfd1e46e9ace7ebf7b118df43_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="b1da085dfd1e46e9ace7ebf7b118df43_2_1-choice_0-label" for="input_b1da085dfd1e46e9ace7ebf7b118df43_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_b1da085dfd1e46e9ace7ebf7b118df43_2_1"> Yes
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<input type="radio" name="input_b1da085dfd1e46e9ace7ebf7b118df43_2_1" id="input_b1da085dfd1e46e9ace7ebf7b118df43_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="b1da085dfd1e46e9ace7ebf7b118df43_2_1-choice_1-label" for="input_b1da085dfd1e46e9ace7ebf7b118df43_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_b1da085dfd1e46e9ace7ebf7b118df43_2_1"> No
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<span id="answer_b1da085dfd1e46e9ace7ebf7b118df43_2_1"/>
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<p>If so, what is it? (If you answered 'No' above, enter '0')</p>
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<h3 class="hd hd-3 problem-header" id="e3e3fd300cb24f80a5a67f0be41058f4-problem-title" aria-describedby="block-v1:MITx+24.118x+2T2020+type@problem+block@e3e3fd300cb24f80a5a67f0be41058f4-problem-progress" tabindex="-1">
Problem 3
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<p>Fool has infinitely many dollar bills, and has labeled each of them with a different natural number (its 'serial number'). One minute before midnight, Fool gives you a dollar bill. Half a minute later, he gives you two dollars. Fifteen seconds later, he gives you four dollars. And so forth. (For each \(i \geq 0\), Fool gives you \(2^i\) dollars \(2^{-i}\) minutes before midnight) There is, however, a catch. Each time you receive money from Fool, you are required to put together all your dollar bills and burn the one with the lowest serial number.</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_e3e3fd300cb24f80a5a67f0be41058f4_2_1" class="inputtype formulaequationinput">
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\(\)
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<p>Next, imagine that you give Fool one dollar at one minute before midnight. Half a minute later, he gives you two dollars. Fifteen seconds later, you give Fool one dollar. Seven and a half seconds later, Fool gives you four dollars. And so forth. (In general, for each \(i \geq 0\), you give Fool one dollar at \(2^{-2i}\) minutes before midnight and Fool gives you \(2^{i+1}\) dollars at \(2^{-(2i+1)}\) minutes before midnight.)</p>
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<legend id="e3e3fd300cb24f80a5a67f0be41058f4_3_1-legend" class="response-fieldset-legend field-group-hd">Assume that, at midnight, you have every dollar bill that you received from Fool and did not return. How much money will you have at midnight?</legend>
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Problem 4
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>You and I each toss a fair coin, and are asked to guess how the other person's coin landed. (Neither of us has any information about how the other's coin landed.) If at least one of us guesses correctly we both get a prize, otherwise we get nothing.</p>
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