In a four‑digit number, the sum of the digits is 10. All the digits are different. What is the largest such four‑digit number?

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?

607

724

?

There are 4! = 24 ways to rank four objects. However, a friend told me that if ties are allowed, the number increases to 75.

I 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.

What 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. 

How many different six-unit paths are there from A to C?

The length and width of a rectangle are whole numbers of centimeters. Neither is divisible by 6. The area of the rectangle is 36 square centimeters. What is the perimeter of the rectangle, in centimeters?

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?

What is the smallest positive number with exactly ten positive integer divisors?

And what is the next one after that?

 777² 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?

Would you rather work seven days at $20 per day or be paid $2 the first day and have your salary double every day for a week?

There are 5 houses on a street: house A, B, C, D and E. The distance between any two adjacent houses is 100 feet.  There are 2 children living in house A, 3 children living in house B, 4 children living in house C, 5 children living in house D and 6 children living in house E.   If the school bus can only make one stop on that street, in front of which house should the bus stop so that the sum of walking distance among all children will be the least?

Jada has 1 penny, 2 nickels, and 1 dime. How many different sums of money can she make, if she uses at least one coin?

If the sum of three numbers equals zero, and the sum of their cubes equals 90, what is their product?

What is the smallest integer n > 1 for which 3ⁿ > n⁹?

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 follows.

Note 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.

Students in Mrs. Walker’s classroom had an estimation contest.  The student whose estimate is the closest to the number of marbles in a jar wins the contest. Vicki, who estimated 135 marbles, won the contest. Timothy, who estimated 150 marbles, got second place. Lyon, who estimated 152, got third place. And Quinn, who estimated 131, got fourth place. What is the exact number of marbles in the jar?

Three lines cut rectangle ABCD into six congruent (identical) squares. 

The perimeter of rectangle ABCD is 30 cm. What is the area of the shaded region, in square centimeters?

Suppose you found an old roll of 15¢ stamps. Can you use a combination of 33¢ stamps and 15¢ stamps to mail a package for exactly $1.77?

How many different color patterns can be created by placing the circles onto the 4 × 4 grid such that each circle is placed on a square with the same number?