MATH PUZZLES BY MAX MILLARD
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I love to make up math puzzles. I never consciously write them, but whenever something occurs in my daily life that makes me to stop and ponder the math, I take the first opportunity to work out the problem and devise a puzzle. The result is a collection of math puzzles ranging from quite basic to fairly complex, whose solutions depend more on creative thinking than pure mathematics. Here are 10 of my favorites. The easiest ones appear first.

EXACTLY HALF

This is the first math puzzle I ever invented. It popped into my head on the morning of my 29th birthday, during a visit to my parents' home in Maine, as I lay in bed and heard my father in the adjacent kitchen. His birthday would be the next day.

Problem: If I turned 29 on December 4, and my father turned 58 on December 5, on which day was I exactly half his age? Do not use leap years in the calculation.

Hint: This problem requires simple algebra.

DAYS OF THE WEEK

You need to find out which day of the week is June 10, 2007. You have only a 2006 calendar. How do you use it to find out which day of the week is June 10, 2007?

BEST DEAL FOR NAPKINS

There are two brands of napkins in the supermarket. Brand 1 costs $2.50 for 360 napkins. Brand 2 costs $3.50 for 500 napkins. By using the calculator just once, how can you determine the better deal?

JUPITER AND SATURN

Jupiter takes 12 years to orbit the sun and Saturn takes 30 years. If they are moving in the same plane and in the same direction, how often does Jupiter pass Saturn?

AROUND THE SKATING RINK

A man and his son Carl go to the Yerba Buena Ice Skating Center. It takes dad 50 seconds to skate all the way around the rink. It takes Carl 1 minute and 25 seconds.

If dad and Carl start at the same point and the same time, dad will immediately take the lead. But at what point will dad pass Carl again? Choose one of the following:

1. Before dad has gone around twice.

2. Just before dad has gone around three times.

3. Just before dad has gone around two and a half times.

COFFEE OR TEA?

Question:

It takes 2 minutes and 30 seconds to heat a cup of coffee or a cup of tea in the microwave by itself, but 4 minutes and 30 seconds to heat them both together, because they each absorb a certain amount of heat.

Question: If I put the coffee in first, and heat it for 1 minute, then add the cup of tea, when do I take each of them out so that they are the right temperature? The answer must be exact.

RIGHT ANGLES

At 9 o'clock, the hands of a clock form a right angle. When is the next time they'll form a right angle? Calculate to the closest second.

CUTTING PAPER IN HALF

You have a piece of paper that measures 8 by 10 inches. You want to be able to trim the paper so that when you cut it in half, then cut the halves in half again, and keep cutting as long as you can, every piece will have the same proportion as the original piece.

For a piece of paper that is 8 inches wide and 10 inches long, the proportion of width to length is .8. If you cut it in half, the proportion of the new pieces will be 5/8 = .625.

Problem: If you keep the paper 10 inches long and trim off some of the 8-inch width, how wide will it be when the proportions of width to length will always be constant, no matter how many times you halve the paper?

HALF A BUBBLE

If you're blowing bubbles in the bathtub, you'll notice that when a bubble hits the water, it immediately changes from a sphere to a half sphere. What is the proportion of the diameter of the whole bubble to that of the half bubble? Use this formula: volume = pi r cubed. You may use a calculator if necessary.

SPACE BETWEEN THE COINS

Place seven round coins in a circle, with one in the middle and the other six on the outside.

Problem: If each coin has a diameter of 1 inch, what is the area of the space between the coins? Use the value of 3.14 for pi.

Approach: If you draw a line from the center of each coin to the coins on either side of it, you will form a hexagon. The distance from the center of the hexagon to each of the most distant points on the outside is 1". Calculate the area of the hexagon, then subtract the area of the circles inside the hexagon. This will tell you the area of the space between the coins.

EXACTLY HALF

Answer: December 3.

Proof: years = y. On December 4 my age was 29y. On December 5 my father's age was 58y. On December 3 I was 29y minus 1 and he was 58y minus 2.

Equation: (29y-1) x 2 = 58y-2

DAYS OF THE WEEK

Answer: Sunday

Solution: Check which day of the week is December 31, 2006. You will find that it is a Sunday. Therefore, you know that January 1, 2007 will be a Monday.

Check which day of the week is January 1, 2006. You'll find that it's a Sunday.

Therefore, the dates for 2007 will be one day of the week later than in 2006.

Check the calendar for June 10, 2006. It's a Saturday. That means June 10, 2007 is a Sunday.

BEST DEAL FOR NAPKINS

Answer: Brand 2 costs $7 for 1000 napkins, which is .7 cents per napkin.

Use the calculator to find out the cost per napkin for Brand 1: $2.50/360 = .694 cents per napkin, which makes Brand 1 the better deal.

JUPITER AND SATURN

Answer: In 300 years, Jupiter will orbit the sun 25 times and Saturn 10 times.

Jupiter will pass Saturn 15 times in 300 years, or once every 20 years.

300/15 = 20 years

AROUND THE SKATING RINK

Answer: 3

Proof: After 50 seconds, dad will have gone around once. By that time, Carl will be 50/85 of the way around, or almost 2/3, so he will be far ahead of dad.

After 100 seconds, dad will have gone around twice. By that time, Carl will have gone around once plus 15/85 of the way, or about 1/6 of the way around. Carl will still be ahead of dad.

By the time dad has gone the rank two and a half times, 125 seconds will have elapsed.

For Carl to go around the rink one and a half times, it will take him 85 x 1.5 = 127.5 seconds. Therefore, dad will pass Carl shortly before both of them are halfway around the rink.

COFFEE OR TEA?

Answer:

Each beverage needs 2.5 minutes of full heat. But when there are two cups in the microwave, they each get 2.5/4.5 of full heat, or 5/9.

For the coffee to get an additional 1.5 minutes of full heat, it would need 9/5 x 1.5, or 2.7 minutes. Multiply .7 by 60 seconds, and you get 42 seconds. So the answer is 2 minutes and 42 seconds.

The tea will need to remain in the microwave for one additional minute.

RIGHT ANGLES

Answer: 9:32:44.

Proof: If you turn the hands of a watch starting at 9 o'clock, you'll see that the hands form a right angle 11 times up to 3 o'clock. That means it makes a right angle every:

360 minutes divided by 11 = 32.727273 minutes.

.727273 minutes x 60 seconds = 43.63638 seconds.

CUTTING PAPER IN HALF

Answer: the square root of 50, or 7.0710678 inches.

proof:

7.0710678 divided by 10 = .70710678

5 divided by 7.0710678 = .70710678

This will work for any piece of paper in which one side is .70710678 of the length of the other side. This means that .70710678, or the square root of .5, is a sort of magic formula. I don't know why it works. I calculated it by gradually getting closer and closer to the answer this way, using the calculator:

width length half length width/length 1/2 length ÷ width8 10 5 .8 .625 7 10 5 .7 .714 7.1 10 5 .71 .704 7.05 10 5 .705 .7092 7.06 10 5 .706 .7082 7.07 10 5 .707 .7072

At this point, I squared 7.07 and found that it was 49.9849. Then, guessing that the answer might be the square root of 50, I calculated that square root, and found that it was exactly right.

Other proof:

x/10 = 5/x

10x/10 = 50/x

x = 50/x

x squared = 50

x = the square root of 50

HALF A BUBBLE

Answer: the cube root of 2, or approximately 1.26.

Proof:

If the original bubble has a radius of 1 inch, it has a volume of 4/3 pi, or about 4.18 cubic inches (4/3 x 3.14 x 1).

If a half sphere had that volume, a whole large sphere would have a volume of 8.36 cubic inches.

For the large sphere:

4/3 pi r cubed = 8.36

r cubed = 8.36 divided by 4/3 pi, or 8.36/4.18, which equals 2.

If the radius cubed is 2, the radius of the half sphere must be the cube root of 2.

This formula will remain constant, regardless of the size of the bubble.

SPACE BETWEEN THE COINS

Solution: The six triangles that make up the hexagon are equalateral triangles, with all three sides measuring 1".

To find the shortest distance from the center of the hexagon to the outside, use this formula for a right angle:

A is the shortest side, B is the second shortest side, C is the diagonal (longest side)

A squared + B squared = C squared.

To calculate the area of the triangle, we need to find the length of B.

A = 1/2, C = 1, so 1/4 + B squared = 1. Therefore B = the square root of .75, or .866

The area of each equalateral triangle = .866 x 1/2 = .433 square inches. The area of all six triangles making up the hexagon is 2.598 square inches.

Inside the hexagon are one whole circle and six 1/3 circles, for a total of 3 circles. The area of each circle is pi x 1/4, or .785 square inches. The area of three circles is 2.355 square inches.

The area between the circles = 2.598 -- 2.355 = .243 square inches.

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