The odometer of the family car shows 15,951 miles. The driver noticed that this number is palindromic: it reads the same backward as forward.
"Curious," the driver said to himself. "It will be a long time before the happens again."
But 2 hours later, the odometer showed a new palindromic number.
How fast was the car travelling in those 2 hours?
This problem appears in The Moscow Puzzles, by Boris A. Kordemsky.
How many plus signs should we put between the digits
to get a total of 99, and where?
(For those who get hooked on this problem: There are two solutions. To find even one is not easy, but the experience will help you put plus signs within
to get a total of 100.)
This problem appears in The Moscow Puzzles, by Boris A.
Kordemsky.
Suppose we have a rectangular notch 30 mm high and abitrarily long; we also have a rectangular peg which is 20 mm wide and 25 mm high. We would like to wedge the peg into the notch so that opposite corners of the peg touch the top and bottom of the notch (see the picture below).
What is the angle between the peg and the notch?
Thanks Marc Frantz for suggesting this lovely problem to me, and for explaining how it relates to wiring braces.
In honor of the end of Daylight Savings Time, we offer the following question:
see a solution here.
This problem originally appeared in a Russian mathematics book. Even reading the problem out loud is not easy!
Three workers can do a piece of work in certain times, viz. A once in 3 weeks, B thrice in 8 weeks, and C five times in 12 weeks. It is desired to know in what time they can finish it jointly.
A pedestrian who had a hat and a stick in his hands was walking home upstream along the side of the river with a speed which was one and a half times greater than the speed of the current. While walking, he threw his hat into the river. "Oh no!" he said to himself a little while later, "I meant to throw the stick in!". He promptly threw the stick in the water and turned around. He ran back towards his hat with a speed which was twice that with which he had walked before. As soon as he caught up with his hat, he plucked it out of the water and turned to walk in the same direction and with the same speed as before. 40 seconds after he got his hat, he passed the stick, which was floating downstream.
How much earlier would he have gotten home if he had not mixed up his hat with the stick?
This problem comes from Mathematical Olympiads by Correspondence.
An advertisement appearing in magazines shows a car stopped at the very edge of the Grand Canyon, and next to that car a set of tire tracks that zoom off of the edge. A bubble above the car says,
and a bubble pointing over the cliff says,
If we assume that the car deccelerates at a constant rate, how much time does it take for the Grand Cherokee to stop?
Thanks to Bill Tyndall for passing this along!
Thanks, Roger!
Every morning for a week, a man climbs the stairs to his office and counts as he goes.
On Sunday, he counts the steps by 2's and has 1 step left over at the top.
On Monday, he counts the steps by 3's and has 2 steps left over at the top.
On Tuesday, he counts the steps by 4's and has 3 steps left over at the top.
On Wednesday, he counts the steps by 5's and has 4 steps left over at the top.
On Thursday, he counts the steps by 6's and has 5 steps left over at the top.
On Friday, he counts the steps by 7's and has 6 steps left over at the top.
On Saturday, he counts the steps 1-by-1. How many stairs are there?
Thanks to Homer Brown, engineer and mathophile, for suggesting this problem to me!
Say this figure shows a portion of a complicated curved line that is completely closed, meaning that the ends meet like a huge rubber band. Some areas are inside the closed region, and others are outside of it. But you can see only the enclosed portion. The rest of the curve is unknown to you. An O marks a spot that is on the outside. Where is the spot marked X? Is it on the inside or the outside?
The Last Problem of the Week for the year ends with the most famous of mathematical problems. Fermat's Last Theorem, first stated some 350 years ago and proved within the last 5 years, says that if n>2, then there are no positive integer solutions to the equation
Either prove this theorem (heh-heh), or find any positive integer solutions to
The famous "Four Color Theorem" says that any flat map can be colored with just four colors. The rules for coloring are
(1) if two areas of a map have an edge or part of an edge in common, then they must be colored with different colors;
(2) any number of colors may meet at a corner; and
(3) we don't require that non-adjacent have to be the same color. (In a real map, for example, you'd want all water-bodies to be blue, and the two pieces of Michigan to be the same color. This is more of a restriction than is allowed in our mathematical map).
Show that four colors is not enough for a torus (the surface of a doughnut). That is, show that there is a 'map' that can be drawn on a torus that needs at least five colors using the rules above.
You are given six balls: two red, two blue, and two green. From each pair of the same color, one is light and one is heavy (it is, however, impossible to tell that without weighing them). All the light balls are the same weight and so are all the heavy ones.
You are also given a balance scale. Can you, using only two weighings, determine which ball is which? If not, can you prove it is impossible to determine that with so few weighings?
This problem was sent in by Kiril
Selverov, '95
The famous English puzzlist Henry Ernest Dudeney (1857--1930) devised this wonderful puzzle.
Take a sheet of paper, crease it into 8 pieces -- 4 across and 2 down -- and number it as shown:
Now find a way to fold it to the size of one of the creased squares, so that the numbers in each of the eight squares are in ascending numberical order, with the number 1 on top.
Fifteen people decide to march every morning for a whole week. Each morning, they will march in 5 rows of 3 people each; but they want to do so in a way so that no two people share a row more than once.
(So, for example, if Andy, Barb, and Carl share a row Sunday, then Andy is never again in a row with either Barb or Carl, and Barb and Carl aren't ever in a row together again either).
Is this marching configuration possible?
FINE PRINT: * This problem reads like a page of of "Who's who in Mathematics". It was first investigated by Kirkland in 1850. It is an example of a problem from Ramsey theory, an area of mathematics which Erdos studied.
When I was a child, my father built me a clubhouse which had a jungle gym roof shaped like a hyperbolic parabaloid. The hyperbolic parabaloid is a popular surface among architects (and in this case physicists), because it is a curved space which is full of straight lines. Indeed, a pair of such lines can be found through any point on the surface.
Find a pair of lines lying in the hyperbolic parabaloid
passing through the point (1, 3, 3). Remember that a line in 3-space is written parametically as
and that there are many ways to write the same line.
Thanks, Dad!
There are only three regular polygons which can tile the plane: equilateral triangles, squares, and equilateral hexagons. Examples of square tilings are easy to find in everyday life. They're on bathroom floors, in windows, in ceiling tiles, etc.
Find 3 examples of hexagonal or triangular tilings used in public places at Franklin & Marshall College.
Marc is standing inside his house, looking out of the window at a 10 foot cube, made of metal bars. On the window are markings that exactly line up with the cube in the yard (at least from where Marc is standing). The markings make a drawing of a cube in perspective on the window.
In the drawing on the window, the front face of the cube measures 2 feet in width and height, and the back face measures 10/7 feet in width and height.
How far is Marc standing from the window? How far is Marc standing from the real cube?
Thanks to Marc Franz for teaching me how to do this!
It was Ramanujan who first discovered that 1729 is the smallest number that can be written as the sum of two cubes two different ways. That is,
Find at least two numbers which can be written as the sum of two squares in two different ways. (By 'square', we mean A2, where A is a positive integer).
Rank in order from smallest to largest:
¥ the number of U.S. Zip-Codes (under the 5+4 system)
¥ the number of English postal codes, and
¥ the number of U.S. telephone numbers under the old system.
In England, (most) postal codes (i.e., Zip Codes) consist of two letters followed by two digits followed by two letters. Frequently, when one telephones a business, the person on the line asks for your postal code and, on entering in the computer, asks "What is your address on Eden St.?" So the code identifies someone down to a block. (Eden St. is one block long.)
Telephone numbers in the U.S. are ten digits long; under the old system they had a three digit area code, of which the first digit could be anything except 0 or 1, the second digit had to be 0 or 1, and the third digit could be any number. The area code is followed by the local number, of which neither of the first two digits could be 0 or 1.
Thanks to George Rosenstein for the tip!
From the 1999 Farmer's Almanac:
Place the numbers 1, 2, 3, 4, 5, 6, 7, 8 into the '#' on the grid so that consecutive numbers do not "touch" each other:
For example, the following combination does not work, because 6 and 7 touch:
Thanks to Mary Ellen Wilson for suggesting this.
There is only one pair of positive integers A and B so that the limit
What is this pair of integers?
Thanks to Kevin Charlwood on the NExT list for suggesting this problem.
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