**dan's math@home - problem of the week - archives****Problem Archives**page 9**Problems Only.**For answers & winners click here.**1-10****. 11-20 . 21-30 . 31-40 . 41-50 . 51-60 . 61-70 . 71-80 . 81-90 . 91-100****101-110****. 111-120 . 121-130 . 131-140 . 141-150 . 151+ .**prob index81 Change 4A Dollar 82 - - Solve Or Die ! 83:The Tree Amigos 84 Crazy Ma'ticians? 85 - - Total-ly Equal! 86- Number of Clues 87: 2 Balls and a Box 88- A Growing Road 89-Gear's The Thing 90 - Cross Numbers! - Problem #81 - Posted Friday, May 12, 2000
- Change For A Dollar? (back to top)
- I went into a store to get change for a dollar.
- The cashier asked, "How do you want that, 100 pennies, or what?
- There are 292 ways involving pennies, nickels, dimes, quarters, or halves."
- "Don't give me any pennies," I said. "That'll cut down the number of ways, eh?"
- a) Just how many ways does this leave? . b) What are all the ways (with no pennies)?
- c) What's the probability I'll get exactly one half-dollar coin in my change?
- List them clearly, like "10 n
+ 5 d," etc. Assume
all ways equally likely.
**Note:**penny:1c, nick:5c, dime:10c, quar:25c, half:50c. - Problem #82 - Posted Thursday, May 25, 2000
- Solve Or Die! (back to top)
- . A man has two medications that he has to take: Med A and Med B. He must take one of each every morning and every evening; any deviation from the prescribed amount will result in death. Each pill costs $1000. The two meds look exactly alike: neither have any writing on them and they weigh exactly the same amount. Once the pills are outside of the bottle, there's no way anyone can tell them apart.
- . One morning the man shook one pill out of the Med A bottle, into his hand. He recapped the bottle, and picked up the Med B bottle. When trying to shake one pill out of the Med B bottle, two fell into his hand. The man now has one Med A and two Med B pills in the same hand and can't tell them apart.
- . What can the man do? He is too poor to throw away 3 pills at $1000 each, and if he takes them incorrectly, or not at all, he will die.
- Problem #83 - Posted Sunday, June 4, 2000
- The Tree Amigos! (back to top)
- I like fruit trees; I have ten in my big flat backyard.
- The trees form five straight lines of four trees each!
- Is this actually possible? If so, how? If not, why not?
- If yes, give (x,y) coords of your trees, or describe exactly how they are placed.
- If it's not possible, try to prove or at least explain why not.

- Problem #84 - Posted Sunday, June 11, 2000
- Crazy Mathematicians? (back to top)
- All mathematicians are either pure or applied. All mathematicians ('mats') are either sane or insane.
- Pure mats always tell the truth about their beliefs; applied mats always lie about their beliefs.
- Sane mathematicians' beliefs are correct; insane mathematicians' beliefs are incorrect.
- Alice says: "I am insane" and "Charlie is pure." Bob says: "I am pure" and "Dorothy is insane."
- Charlie says: "I am applied" and "Bob is applied." Dorothy says: "I am sane" and "Charlie is sane."
- Describe each mathematician as Pure or Applied, Sane or Insane; explain your reasoning!
- Problem #85 - Posted Friday, June 23, 2000
- Total-ly Equal ! (back to top)
- Each icon in the table represents a real number.
- Some of the sums are given on the outside.
- The total of all five icons is a 2-digit factorial.
- It's up to you to find the value of each icon,
- and figure out the totals x, y, z, and w.
- Show your steps and reasoning; be clear!

**=****20****=****17****=****35****=****x****||****||****||****||****\\****y****19****18****z****w**- Problem #86 - Posted Saturday, July 1, 2000
- A Number of Clues (back to top)
- (Ok, I admit it -- I stole this problem from one of my many e-mail fans. Thanks!)
- I have eight integers in my list. Use the clues to find my numbers.
- 1. The range (statistical sense) of
the numbers in my list is 93.

2. One of my numbers is an abundant number between 21 and 25.

3. One of the numbers in my list is a perfect cube less than 10.

4. The arithmetic mean of the numbers in my list is 40.

5. The mode of the list is the sum of the first two perfect numbers.

6. The median of the numbers in my list is 29.5.

7. The largest number in my list is the smallest 3-digit palindrome.

8. The sum of the digits of one of the numbers is its square root. - What are the eight numbers? . . Show your steps and reasoning.
- Problem #87 - Posted Thursday, July 13, 2000
- Two Balls and a Box (back to top)
- A perfect cube is inscribed in a big sphere, and then another (smaller) sphere is inscribed in the cube.
- a. What is the ratio of the
**radii**of the two spheres?* b. What's the ratio of the**volumes**of the two - spheres?* c. Which is bigger, the ratio of the volume of the big ball to the vol of the box, or the vol
- of box to the vol of small ball?
- * Answers to a and b should be more than 1. Show your steps and reasoning; be clear!
- Problem #88 - Posted Monday, July 24, 2000
- The Long and Growing Road (back to top)
- Tenny the Elephant starts along a 10-mile long road and walks one mile per day. Each night the road lengthens by 10 miles, proportionally on each side, so beginning the second day there are 2 miles behind Tenny and 18 miles in front. At the end of that day he'll have 3 miles behind and 17 miles in front, and that night the road will (proportionally) get another 10 miles longer. If he walks one mile each day and the road gets 10 miles longer each night, will he ever reach the end of the road? If not, why not? If so, how many days will it take? Show your steps and reasoning; be clear!
- Problem #89 - Posted Thursday, August 3, 2000
- Gear's The Thing... (back to top)
- Lants Headstrong pedals at the rate of 105 revolutions per minute. His bike chain connects a 53-tooth sprocket on the front (pedal) ring and a 12-tooth sprocket on the back wheel. If the diameter of his (back) wheel is 27 inches, how many times will his left leg push down in a 43-km time trial, and what will his (winning) time be (in minutes and seconds)? Show your steps and reasoning; be clear!

Problem #90 - Posted Sunday, August 13, 2000

Cross-Numbers ! (back to top)

Fit a bunch of the words ONE, TWO, ... , TEN, all connected crossword-style, at most once each, into a 6x6 grid. The sample shown has 'sum 6' because 1 + 2 + 3 = 6. The winner is the one with the largest sum before the deadline; ties will be broken by largest number of numbers, then most total letters in grid, then entry date. Answers, showing sum, and 'row form' using '*' for blank. Row form for entry shown is {******,*TWO**,*H*N**,*R*E**,*E****,*E****}.

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