Place a number in each of the following empty boxes so that the sum
of the numbers in any 3 consecutive boxes is 2013. What is the number
that should go in the box with the question mark?
There are 4! = 24 ways to rank four objects. However, a friend told me that if ties
are allowed, the number increases to 75.
attempted to list all the possibilities by first listing the 24 orderings of
four objects, then using brackets to group ties involving two players, then
group ties involving three players, and finally the single case in which all
four objects are tied. But something has gone wrong; my list includes just 69
possibilities, not 75.
happened? Did I miss something, or was my friend mistaken?
The rectangle shown consists of eight squares. The length of each side of each
square is 1 unit. The length of the shortest path from A to C using the lines
shown is 6 units.
many different six-unit paths are there from A to C?
A calendar year is typically referred to as a four‑digit number, as in 2008, or
as a two‑digit number, as in ’08. Sometimes, the two‑digit number divides
evenly into the four‑digit number, with no remainder.
How many times did this happen during the twentieth century?
is the smallest positive number with exactly ten positive integer divisors?
And what is the next one after that?
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?
Write the number 41 in a box. Now move in a counterclockwise direction, creating new
boxes and each time adding 1 to the number inside. This spiral starts out as
that the numbers in bold — 41, 43, and 47 — are prime numbers
(numbers whose only divisors are themselves and 1), and they occur along a diagonal.
If you keep filling in the spiral, what is the first number that is not prime to
appear along this diagonal?
The letters of EAT can be
rearranged to become TEA by moving the third letter (T) to the first
position and by moving the other letters one position to the right. This
process could be described as 1→2, 2→3, 3→1.
When this same process is applied again, then TEA becomes ATE.
Similarly, the process 1→3, 2→2, 3→1, 4→4
will convert TONE to NOTE. When the process is applied again, NOTE returns to
TONE. (Not very interesting, is it?)
Your challenge begins with
the five letters A, E, M, S, and T. Use them to form a common English word.
Then, rearrange the letters to form a second common English word. Finally,
apply the same process of rearrangement to form a third common English word.
Can you do it?
Label the ten points in the grid shown with the letters A-J so that
AB < BC < CD < … < HI < IJ.
Three lines cut rectangle ABCD into six congruent
The perimeter of rectangle ABCD is 30
cm. What is the area of the shaded region, in square centimeters?