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Ask The Wizard #378

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?

Gialmere

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.

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?

anonymous

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

Barbara first: 7.5 minutes
Chrissy first: 6 minutes

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?

anonymous

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.