Q: A farmer has a 1x1 square piece of land. He wishes to divide it equally among his three children. He may use no more than three straight sections of fencing to divide the property. How should he do it to minimize the length of the fencing?

<p><div><button onclick='var btn = this; var div = btn.parentNode.childNodes[1]; if (div.style.display != "block") { div.style.display="block"; } else { div.style.display="none"; }' style='padding:2px 4px;'>Answer</button><div class='spoiler'>2/3 + sqrt(15)/4 =~ 1.634912503219 </div></div> </p> <p>Here is my <a href="/wizfiles/pdf/ask-the-wizard-378.pdf">soltion</a> (PDF). </p> <P>This question is asked and discussed in my forum at <a href="https://wizardofvegas.com/forum/questions-and-answers/math/34502-easy-math-puzzles/58#post853635" target=_blank>Wizard of Vegas</a>. </p>

Q: <p>Amy, Barbara, and Chrissy decide to play a game of tag. They all start in the same place on an open field. Their running speeds, in feet per minute, are: </p> <ul> <li>Amy: 500 </li> <li>Barbara: 400 </li> <li>Chrissy: 300 </li> </ul> <P>Amy is appointed to be "it" first. Barbara and Chrissy take off, in opposite directions, which they both maintain, while Amy does a one-minute countdown. </p> <p>Who should Amy go after first, to minimize the total time required to tag both Barbara and Chrissy? For extra credit, what is the time required both ways? </p>

<div><button onclick='var btn = this; var div = btn.parentNode.childNodes[1]; if (div.style.display != "block") { div.style.display="block"; } else { div.style.display="none"; }' style='padding:2px 4px;'>Answer</button><div class='spoiler'><p>She should go after Chrissy first. Here is the time required both ways: </p> Barbara first: 7.5 minutes<br> Chrissy first: 6 minutes</p></div></div>

Question 1

A farmer has a 1x1 square piece of land. He wishes to divide it equally among his three children. He may use no more than three straight sections of fencing to divide the property. How should he do it to minimize the length of the fencing?

Accepted Answer

2/3 + sqrt(15)/4 =~ 1.634912503219

Here is my soltion (PDF).

This question is asked and discussed in my forum at Wizard of Vegas.

Question 2

Amy, Barbara, and Chrissy decide to play a game of tag. They all start in the same place on an open field. Their running speeds, in feet per minute, are:

Amy: 500
Barbara: 400
Chrissy: 300

Amy is appointed to be "it" first. Barbara and Chrissy take off, in opposite directions, which they both maintain, while Amy does a one-minute countdown.

Who should Amy go after first, to minimize the total time required to tag both Barbara and Chrissy? For extra credit, what is the time required both ways?

Accepted Answer

She should go after Chrissy first. Here is the time required both ways:

Barbara first: 7.5 minutes
Chrissy first: 6 minutes

Question 3

You have ten icosahedrons, with sides numbered 1 to 20 and roll all of them. What is the mean roll on the die with the highest roll?

Accepted Answer

24,163,571,680,850 / 20¹⁰ =~ 18.640276

The following table shows the number of combinations for any given total. The formula for that is simply n^10 - (n-1)^10. The other two columns show the probability and contribution to the return. The lower right cell shows an average of 18.640276.

Highest side	Combinations	Probability	Return
1	1	0.000000	0.000000
2	1,023	0.000000	0.000000
3	58,025	0.000000	0.000000
4	989,527	0.000000	0.000000
5	8,717,049	0.000001	0.000004
6	50,700,551	0.000005	0.000030
7	222,009,073	0.000022	0.000152
8	791,266,575	0.000077	0.000618
9	2,413,042,577	0.000236	0.002121
10	6,513,215,599	0.000636	0.006361
11	15,937,424,601	0.001556	0.017120
12	35,979,939,623	0.003514	0.042164
13	75,941,127,625	0.007416	0.096410
14	151,396,163,127	0.014785	0.206987
15	287,395,735,649	0.028066	0.420990
16	522,861,237,151	0.051061	0.816971
17	916,482,272,673	0.089500	1.521504
18	1,554,473,326,175	0.151804	2.732473
19	2,560,599,031,177	0.250058	4.751111
20	4,108,933,742,199	0.401263	8.025261
Total	10,240,000,000,000	1.000000	18.640276

I don't know of any formula for the answer that doesn't require a summation sign. If you do, let me know.

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Ask The Wizard #378

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