20/20 Visions

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?

20 Questions

1.

**{ EASY }** Someone is thinking of a whole number between 1 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.

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?

**Solution** Want to see the

solution to this puzzle?

Got new solutions for the puzzle? Want to see other people's solutions? Talk to the puzzle master in his discussion forum at

www.scottkim.com.

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