Problems for Solution
- Using as many of the digits 0 through 9 as possible, but not more than once each, form a set of prime numbers whose sum is both a prime and a minimum.
- Using each of the digits 0 through 9 exactly once, construct a set of prime numbers whose product is a minimum.
- I recently refinanced the mortgage on my home. I obtained a 15 year mortgage at 7.25% with no points and a $1551.87 monthly payment. Before I made the first payment, however, I received a raise of $147.18 per month at my job. I decided to send the amount of my raise to my lender, in addition to my required monthly payment, so that my loan will be paid off a little early. For how long will I now be making payments on this loan?
- Construct a re-entrant knights tour so that the sum of the entries:
- on the border of the chessboard is a minimum.
- on the diagonal of the chessboard is a minimum.
- on a row of the chessboard is a minimum.
- For each digit, n, from 1 to 9, find the smallest number, m, such that the repeating decimal representation of the reciprocal of m contains all digits except n.
- For each digit, n, from 0 to 8, find the smallest number, m, such that the repeating decimal representation of m / (m + 1) contains all digits except n.
- Prove that 10n + 1 is always factorable when n is greater than two, or provide a counterexample.
- I have several boxes each containing the same number of coins (1¢, 5¢, 10¢ and 25¢) worth a total of $1.00 or less. If no two boxes have the same combination of coins and the number of boxes is a multiple of the number of coins in each box, what is the maximum number of coins that I can have and what is their total value?
- Kidnappers often ask for a ransom of small ($1, $5, $10, $20), unmarked bills for the safe return of their hostage. In how many ways can a ransom of $1,000,000 be paid with these small bills?
- A box contains n coins. The box is shaken throughly and all coins that come up heads are discarded. The box is shaken again with the remaining coins, and again all heads are discarded. Find the expected value of the number of times that this shake-and-discard procedure must be performed in order to discard all of the coins.
- A very small flea jumps between two walls, one foot apart. The flea starts at the base of one wall and jumps in a straight line directly towards the other. The length of its first jump is 1/2 ft., the length of its next jump is 1/3 ft., the next one 1/4 ft., the next 1/5 ft. and so on. When the length of the flea's jump is greater than its distance from the wall that is is jumping at, it hits the wall and bounces off with no decrease in the length of its jump. For example, when the flea is 1/6 ft. from the wall and then jumps 1/4 ft., it bounces off the wall and lands 1/12 ft. from the wall.
- Suppose the flea changes direction and jumps towards the opposite wall whenever it bounces off the wall at which it is jumping
- Will the flea ever land at the same point twice?
- Will the flea ever land at the base of either wall?
- Suppose the flea continues to jump towards the first wall, even after bouncing off.
- Will the flea ever land at the same point twice?
- Will the flea ever land at the base of either wall?
- In the decimal system, the product of two consecutive integers, n(n + 1), terminates in 0, 2 or 6; and if it terminates in 6, then it terminates in 06 or 56. Find the smallest palindrome terminating in 56 that is the product of two consecutive integers, or prove that none exists.
E-mail: ewcollins@att.net