20/20 Visions

Scott Kim

To celebrate the 20th anniversary of Discover, put your hand–and mind–to these puzzles, which prominently feature the number 20. Afterward, try to figure out how many times the numeral 20 appears on this page.

20 Trees

1. {EASY} The 14 trees at left are planted in five rows of four trees each. Can you plant just 10 trees and still form five rows of four trees? Rows must be straight, and each row must lie on a different line. There is more than one solution.

2. {MEDIUM} Can you plant 20 trees to form 20 straight rows of four trees each? Hint: Use the previous solution.

3. {HARD} How many rows of five trees each can you make by planting 20 trees?

BOGGLERS SOLUTIONS

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1. The 10 trees form a five-pointed star. For one of many other solutions (right), draw five lines that overlap one another in a different way.

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2. This solution uses three nested five-pointed stars.

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3. One solution uses two nested five-pointed stars at different angles.

20 Questions

1. {EASY} Someone is thinking of a whole number between I and 1,000,000. How can you find out this number by asking 20 yes-no questions?

2. {MEDIUM) Again, someone is thinking of a whole number between 1 and 1,000,000. You can ask questions to which the answer is always one of three choices: a, b, or c. How many questions must you ask to be certain what number he has in mind?

3. {HARD} Now someone is thinking of a whole number between 1 and 100. Suppose you ask the following 20 yes-no questions:

Is the number evenly divisible by 1? Is the number evenly divisible by 2? Is the number evenly divisible by 3? And so on, up to: Is the number evenly divisible by 20?

Based on the answers, you may or may not be able to figure out the number. For instance, you can’t distinguish the numbers 16 and 32 from the answers to these yes-no questions, since both numbers are divisible only by 1, 2, 4, 8, and 16. What numbers are indistinguishable from 27? From 20? 4? 97? 93?

4. {FORGET ABOUT IT} List all the numbers between 1 and 100 that you can deduce with certainty from the answers to the 20 questions described in Problem 3.

BOGGLERS SOLUTIONS

1. Your first question should be “Is your number greater than 500,000?” Based on the answer to that question, keep asking questions that divide the remaining range of possible answers in half. For instance, the next question will be either “Is your number greater than 250,000?” or “Is your number greater than 750,000?” depending on whether the answer to the first question is no or yes. Each question doubles your chances of guessing the right number, and [2.sup.20] is 1,048,576–more than enough chances to deduce any number from 1 to 1,000,000.

2. As in the previous solution, you should ask questions that help limit the range of possible answers. For example, your first question should be: “Is the number (a) less than 333,333; (b) between 333,334 and 666,666; or (c) between 666,667 and 1,000,000?” Each successive multiple-choice question you ask triples your chances of getting the answer. Thirteen questions are needed, because the first power of 3 that is at least as large as 1,000,000 is [3.sup.13], which is 1,594,323.

3. 27 is indistinguishable from 9 and 81. 20 is indistinguishable from 100. Four is indistinguishable from 92. 97 is indistinguishable from prime numbers greater than 23, and 93 is indistinguishable from 87 and 69.

4. The deducible numbers up to 100 are: 6, 8, 11, 12, 13, 17, 19, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 51, 52, 55, 56, 57 60, 63, 65, 66, 68, 70, 72, 76, 77, 78, 80, 84, 85, 88, 90, 95, 99.

20 Painted Houses

1. {EASY} The city of Octopolis has eight houses arranged as shown above. It takes one day to paint a house and one more day for the paint to dry. Local laws forbid painting a house that is adjacent horizontally, vertically, or diagonally to a house whose paint is still wet, because the paint fumes are too strong. Can you figure out an order in which you can paint all eight houses in eight days? Number the houses from 1 to 8 in the order in which you would paint them.

2. {MEDIUM} The teeming and rigidly structured city of Hexadecipolis comprises 16 houses arranged in four rows of four. A house can be painted in one day, but it takes two days for the paint to dry. Again, local laws forbid painting a house that is adjacent to a house with wet paint. Can you determine an order that lets you paint all 16 houses in 16 days? Hint: The numbering on adjacent houses must differ by at least 3.

3. {HARD} Hexadecipolis’s sister city, Icosapolis, known as the Humid City, has 20 houses arranged in four rows of five. Houses can still be painted in one day, but now it takes three days for the paint to dry. And, of course, you can’t paint a house adjacent to one that is still wet. Is there an order in which you can paint all of the houses in 20 days? Hint: The numbering on adjacent houses must differ by at least 4.

4. {TRULY BOGGLING} An intense summer has descended upon Icosapolis, and it now takes four days for paint to dry. It’s not the heat; it’s the humidity. Can you figure out how to paint all 20 houses in 21 days?

BOGGLERS SOLUTIONS

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1. This solution is unique, ignoring mere reflections.

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2. This puzzle has many other solutions.

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3. This puzzle has many other solutions.

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4. This solution requires a one-day wait after house number 18.

HOW MANY 2Os?

The numeral 20 appears 20 times on page 106, as follows: 5 times in 20/20 Visions (introductory text). 4 times in 20 Painted Houses 6 times in 20 Questions 4 times in 20 Trees and 1 time in “October 2000”

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