## Problems for Entertainment and Enrichment

Over my years teaching mathematics at many levels, I accumulated a number of quirky puzzles related to math. Most are accessible by simple arithmetic. A number can be used as an introduction to related mathematical topics. I am sharing these as a potential classroom resource.

Two cautions: Regrettably, I did not consistently collect source information nor did I check publication rights. Many of these date back a number of years, and I no longer remember whether or not I’ve verified the solution.

1)

Some people consider the depression year 1930 unlucky because the sum of the digits totaled 13. What is the next year in which the numbers will again total 13? (Otto P. Kramer, *Games Magazine*, Feb. 1983)

[2029]

2)

Do this in your head: Which is larger, 94.1% of 23.25, or 23.25% of 94.1? (Michael Ecker, *Games Magazine*, Sept.-Oct., 1981)

[They are equal by the commutative property: AxB = BxA]

3)

Riddle: Two days ago, I was only twenty-eight. Next year I’ll be thirty-one. What day is my birthday? (*Games Magazine*, Sept.-Oct., 1981)

[December 31. Two days ago, was December 30, and I was 28. Today it is January 1, and I am 29. At then end of this year, when it’s December 31 again, I’ll be 30. But NEXT YEAR, on December 31, I will be 31.]

4)

What are the chances? The race track is terribly dangerous. Drivers must first cross a very narrow bridge that sends one out of every five cars into the water. Next comes a grueling hairpin turn which forces three out of every ten cars into a ravine. Th pitch-dark tunnel that follows is so treacherous that one out of every ten cars never emerges. Last comes a sandy stretch in which two out of every five cars bogs down.

Given these pitfalls, what percentage of those cars competing will make their way successfully to the finish line? (*Mathematical Games*, M. Berrondo)

[The probability of these sequential events is the product of the individual probabilities. Note that the answer asks for SUCCESSFUL finish, but the individual hazards are given in terms of FAILURE. 4/5 x 7/10 x 9/10 x 3/5 = .3024 or about 30%]

5)

Change the link HARD into EASY. Change one letter, starting with the word HARD, to make a new word at each step of the chain ending with EASY. (Five steps)

[HARD —> CARD —> CART —> CAST —> EAST —> EASY]

6)

Place six vertical lines in a row like this: | | | | | | . Now add five straight lines until you have exactly nine.

[NINE]

7)

A corny riddle: A box contains nine ears of corn. A squirrel carries out three ears each day, but it takes him nine days to completely empty the box. How can you logically explain this?

[Two of the three daily “ears” are the squirrel's. The squirrel only removes one ear of corn each day.]

8)

What non capitalized word in the English language is distinguished by three dotted letters in a row? (John B. Klein, * Games Magazine*, March-April, 1982)

[hijinks]

9)

Norma, Naomi, and Nan are engaged to be married. Who will marry whom if: Naomi is not engaged to the artist; Joe is an author; the doctor’s wife is not Nan; Elliot is engaged to Norma; and Matt is the artist? (*Games Magazine*, March-April, 1982)

[Norma <–> Elliot, Nan <–> Matt, Naomi <–> Joe]

10)

Three cards are dealt face down. A diamond lies to the left of a spade, a two is to the right of a jack, a nine lies to the left of a club, and a club is on the left of a spade. What are the cards?

[nine of diamonds, jack of clubs, two of spades]

11)

All of the following words share a common characteristic: Farad, decibel, henry, joule, and tesla. What is this common characteristic? (Sydney Harris)

[All are units of measurement named after the men who devised them: Michail Faraday, Alexander Grahm Bell, Joseph Henry, James Joule, and Nikola Tesla]

12)

If a pen costs a dollar more than an eraser, and together they cost $1.10, what is the price of each. (This problem not adjusted for inflation.)

[If E = Eraser, then E + (E + $1) = 1.10. Solve for E = $0.05, and pencil = $1.05.]

13)

Divide thirty by one half and add ten. What is the answer?

[70. There are two halfs in every whole, so there are sixty one-halfs in thirty. Now, add ten.]

14)

Suppose there are seven amoebas in the bottom of a jar. They are multiplying so fast, they double their volume every minute. If it takes forty minutes for the amoebas to fill half the jar, how much longer will it take to fill the whole jar?

[One minute. The jar is half full. Since they double in volume every minute, they will double that half-full to make the jar completely full in one more minute. This is a classic example of the (scary) idea of exponential growth.]

15)

How many two-digit, positive whole numbers are there?

[90 Easier to work backwards: There are 99 whole numbers from 1 to 99? Nine of those are one-digit numbers ( 1 – 9). So 99 – 9 = 90]

16)

Quick! What two whole numbers multiply together to make 13?

[13 x 1]