A Friedman number is a number that can be
represented with an expression that uses only the digits in the number. In
addition, the expression can include +, –, ×, ÷, exponents and parentheses, but
nothing else. For instance, 25 is a Friedman number because it can be
represented as 52. A nice
Friedman number is a Friedman number for which the digits occur in the same
order in the expression as they do in the original number.
So, 343 is a nice Friedman number, because it can be represented by an
expression with the digits 3, 4, and 3 in the same order:
343 = (3 + 4)3
seven nice Friedman numbers are 127, 343, 736, 1285, 2187, 2502, 2592. Can you
find an expression for each of them?
A 10 × 10 grid is painted
with three primary colors (red, yellow, and blue) and three secondary colors
(green, purple, and orange). The secondary colors are made by mixing equal
parts of the appropriate primary colors — that is, red and yellow are
mixed to make orange, red and blue to make purple, and yellow and blue to make
The figure at left shows
squares that were painted red and blue. No other squares were painted either
red or blue.
Suppose that each small
square requires a quart of paint. Altogether, 31 quarts of red paint, 40 quarts
of blue paint, and 29 quarts of yellow paint were used to paint the entire
10 × 10 grid.
Given this information, can
you determine if there were more yellow or purple squares? And how many more?
Label the ten points in the grid shown with the letters A-J so that
AB < BC < CD < … < HI < IJ.
Look at the panel of
elevator buttons shown. Can you find a set of three buttons whose centers
form the vertices of a right triangle and whose numbers are the side lengths of
a right triangle? (The classic 3-4-5 right triangle doesn’t work, because the
3, 4, and 5 buttons don’t form a right triangle on the elevator panel.)
And after you’ve found one
set, can you find another?
means 777 × 777,
7773 means 777 × 777 × 777,
and so on.
Suppose 7777 is completely multiplied
out. What is the units digit of
the resulting product?
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 that happens again.”
But 2 hours later, the
odometer showed a new palindromic number.
How fast was the car
traveling in those 2 hours
Create two nine-digit numbers, using the digits 1-9 in some order, so
that one can be used as the numerator of a fraction and the other as
the denominator to yield an extremely good approximation of
. Each digit 1-9 will be used exactly twice, once in the numerator and once in the denominator.
How close can you get to the exact value of
In May 1999, two National League baseball players, Joe McEwing of the St.Louis Cardinals and Mike Lieberthal of the Philadelphia Phillies, each had the batting averages, as show below.
Lieberthal: Bats- 132; Hits- 45; Batting Average- 0.341
McEwing: Bats- 132; Hits- 45; Batting Average- 0.341
Suppose McEwing then batted 0.800 (4 hits in 5 at bats), and Lieberthal was perfect (3 hits in 3 at bats). Which player now has the higher batting average? Are you surprised?