**dan's math@home - problem of the week - archives****Problem Archives**page 11**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 index101 -- The Fab Five ! 102 - Similar but Diff 103 - Weird Numbers 104 Dan's Prime Code 105- Ez Come, Ez Go 106- New Year Phone 107- Geometry Gems 108 - Cover The Cube 109- Based On What? 110 - Family Of Four - Problem #101 - Posted Sunday, November 19, 2000
- The Fab Five ! (back to top)
- A set contains five integers. When distinct elements of this set are added together,
- two at a time, the complete list of different possible sums that result is:
- 637, 669, 794, 915, 919, 951, 1040, 1072, 1197.
- a) Figure out (if possible) the original five integers in the set.
- b) What are the means; of the original set, and the set of sums ?
- c) Given a set of 'sums', do five such numbers always exist ?
- Problem #102 - Posted Monday, November 27, 2000
- Similar but Different (back to top)
- Drew drew two similar triangles, both with integer sides. Two sides of one triangle
- were the same as two in the other.The other (unmatching) sides differed by a prime.
- a) What are the smallest triangles making this possible? (Give corresp. prime difference.)
- b) What is the smallest prime for which this is possible? (Give corresponding triangles.)
Answers for a) and b) may or may not be different. - Problem #103 - Posted Friday, December 8, 2000
- Weird Numbers! (back to top)
- You may know a 'perfect number' is one whose proper divisors add up to the number,
- such as 6: 1+2+3 = 6. In an 'abundant number' the divisors add up to more than the
- number, like 12: 1+2+3+4+6 = 16 > 12. A 'weird number' is an abundant number with
- no subset of divisors adding to the number. The number 12 isn't weird 'cause 1+2+3+6 = 12.
- What are the first two weird numbers?
- Problem #104 - Posted Sunday, December 17, 2000
- Dan's Prime CodeTM (back to top)
- The first 26 primes (2, 3, 5, ...) can be put in correspondence with the letters A through
- Z, so the 'Prime Code' for the word CAB would be the product 5 * 2 * 3 = 30.
- a) What's the Prime Code for the word BIKER ?
- b) Decode this message... 913 1511191 110618.
- c) What (English) word comes closest to a million?

- Please submit all parts in one message. No proper nouns. Ranked by order received AND best c); results avgd.
- One point penalty for each resubmission of 'improved' answers to c).
- Problem #105 - Posted Monday, December 25, 2000
- Easy Come, Easy Go! (back to top)
- At their traditional end-of-the-millenium poker game, Clifton and Lawrence agree on
- the stakes for each hand: The loser pays 1/3 of the money he has remaining, to the winner.
- After a while, Lawrence gives up: "You now have exactly three times the cash I have,
- you've won the last few hands, and I've lost just about four bucks!"
- "But you won every hand before that," Clifton replied, "in fact you've won the same
- number of hands I have!" How many hands were played, how much money did they
- each start with, and how much do they have now?
- Problem #106 - Posted Wednesday, January 3, 2001
- New Year, New Phone! (back to top)
- After my recent move, I got a new phone number. I forgot to write it down, but it had a
- 3-digit prefix and the rest was another 4 digits, like this: xxx-xxxx. I did remember that
- the prefix, subtracted from half the square of the rest of the number, gave me the whole
- phone number as a result ! What was my new number ? (Explain steps fully for best ranking!)
- Problem #107 - Posted Friday, January 12, 2001
- Gee, Geometry Gems ! (back to top)
- a) Given that two of the three sides of a right triangle are 3 and 4, what's the shortest possible
- length for the third side? b) If (6,9) and (10,3) are the coordinates of two opposite vertices of
- a square, what are the coords of the other two? c) One circle has radius 5 and center at (0, 5).
- A second circle has radius 12, center (12,0). Find the radius and center of a third circle which
- passes through the center of the 2nd circle and both intersection pts of the first two circles.
- Problem #108 - Posted Sunday, January 21, 2001
- Cover The Cube ! (back to top)
- The T-shape at the right can cover the six faces of a cube. How many other shapes can you find that cover the cube? Please give your answers as lists of the six squares used. The T-shape in the picture would be called {a5, b2, b3, b4, b5, c5}. Rotations and reflections (flips) don't count as different. Shapes must be connected; squares must touch along a whole edge.

- Problem #109 - Posted Tuesday, January 30, 2001
- Based On What?! Here's a fun rhyme I found; see if you can answer it! (back to top)
- The square of nine is 121 ; I know it looks quite weird.
- But still I say it's really true ; the way we figure here.
- And nine times ten is 132 ; the self-same rule, you see.
- So whatcha say I'd have to write for five times twenty-three?
- Problem #110 - Posted Saturday, February 10, 2001
- Family Of Four (There may be more than one answer.) (back to top)
- "Hey, nice looking family!" said Fred, seeing the photo. "I met your younger boy today;
- he said he was nine, and your wife reminded me his brother is older."
- Frank agreed, saying, "It's odd about all our ages. If you total the squares of my age and
- the boys' ages, you get my wife's age times the total of my age and the boys' ages."
- If the ages are all whole numbers, can you figure them out?
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