dan's math@home - problem of the week - archives
 
 
Problem Archives page 20
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 . 151-160 . 161-170 . 171-180
181-190 . 191-200 . 201-210 . 211-220 . 221-230 . 231+ . prob index
 
191 -- The Disk Cutters
192 --- Slice Into Dice !
193 - Even That's Odd !
194 Uncomm'n Multiple
195 - Over-Committeed
196 - The Pentominator
197 CommonRemainder
198 - Odd and Different
199 - Twice The Angle !
200- Run with the Dogs
 
 
Problem #191 - Posted Friday, January 2, 2004
The Disk Cutters (back to top)
From a smooth 30-inch-diameter circular plywood disk, two smaller circular disks
of diameters 20" and 10" are cut. What is the largest circular disk that can be cut
from one (either) piece of the remaining plywood?
Ignore width of sawblade when cutting. Explain your answer carefully.
 
Problem #192 - Posted Sunday, January 18, 2004
Slice Into Dice! (back to top)
I painted a block of wood that's a x b x c cm (a <= b <= c whole numbers).
If I then cut it up into one-cm cubes, half of the cubes have paint
on them and half don't. What are all possible values of a, b, and c?
Ignore width of sawblade when cutting. Explain answer carefully.

 
Problem #193 - Posted Friday, February 6, 2004
Even That's Odd ! (back to top)
In a valid base ten multiplication problem, all the
even digits were replaced by E's, and the odd digits
were written as O's, giving the result at the right.
What must the original problem have been ?
Explain answer carefully.
 
Problem #194 - Posted Thursday, February 26, 2004
An Uncommon Multiple ! (back to top)
Prove that for all integer values of x,
is an exact multiple of the number 8640 . . Explain answer carefully.
 
 
Problem #195 - Posted Saturday, March 6, 2004
Over - Committeed ! (back to top)
Fifteen college teachers (A,B,...,O) must serve on a total of twenty committees (1-20), such that . . .
i) Each teacher is on exactly four committees,
ii) Each committee has three teachers on it,
iii) No two committees have more than one teacher in common.
List (or accurately describe) such a committee structure, or else prove it can't exist. Explain answer carefully.
 
 
Problem #196 - Posted Wednesday, March 17, 2004.
The Pentominator ! (back to top)
"Pentominoes" are objects made from five congruent squares
attached along full edges. The 12 possible shapes are as shown:
Use a simple grid of text with rows like "TTTUUFF..." (no need for attached pictures) to show how
to use all 12 to: a) Make any 6 x 10 rectangle (there are over 1000 solutions), b) Make two 6 x 5
rectangles (using all twelve pieces once!), c) Make into an 8 x 8 square, missing its four corners.
Each shape can be rotated or flipped over at will. Show your answer carefully.
 
 
Problem #197 - Posted Saturday, March 27, 2004
A Common Remainder (back to top)
480608 ,- 508811 , 723217. These three numbers, when divided
by a certain natural number > 1 , all yield the same remainder.
What is that divisor and that remainder? Show your reasoning carefully.
 
 
 
Problem #198 - Posted Friday, April 9, 2004
Odd and Different! (back to top)
 5+1  3+3  3+1+1+1  1+1+1+1+1+1
 6  5+1  4+2  3+2+1
There are four ways to make 6 as a sum of odd numbers,
and four ways to make 6 as a sum of different numbers.
a) Write each number from 1 to 10 in all possible ways (arranged as shown from
largest to smallest) as the sum of odds, sum of distincts, and sum of any.
b) Prove (for general n) that the number of sums of odd numbers giving n is the
number of sums of distinct numbers giving n, or find the first counterexample.
Show your reasoning carefully.
 
 
Problem #199 - Posted Sunday, April 18, 2004
Twice The Angle (back to top)
Many triangles have one angle double, or twice, another (e.g. angles 30,60,90).
a) Find the smallest integral triangle with one angle twice another.
b) Find general expressions that can generate infinitely many such triangles (a, b, c).
In part a), 'smallest' means smallest perimeter a+b+c. 'Integral triangles' have all three sides integers.
Show reasoning carefully. Partial answers accepted.
 
 
Problem #200 - Posted Friday, April 30, 2004
Run With The Dogs (back to top)
The writer Jack London had to go from Skagway to a remote camp in a sled
pulled by 5 husky dogs. For 24 hours the dogs pulled the sled at full speed.
Then 2 dogs ran off with a pack of wolves, and the sled was slowed down
proportionally. Jack reached camp 48 hours later than he planned. "If those
2 huskies had held on for 50 more miles, I would have been only half as late."
How far was the camp from Skagway and how long did it take Jack to get there?
Show reasoning carefully.
 
 
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YOU CAN ALWAYS FIND ME AT dansmath.com - Dan the Man Bach - 2004 A.D.
 
 
Problem Archives Index
 
Probs & answers . 1-10 . 11-20 . 21-30 . 31-40 . 41-50 . 51-60 . 61-70 . 71-80 . 81-90 . 91-100
Problems only . . . 1-10 . 11-20 . 21-30 . 31-40 . 41-50 . 51-60 . 61-70 . 71-80 . 81-90 . 91-100
Probs & answers . 101-110 . 111-120 . 121-130 . 131-140 . 141-150 . 151-160 . 161-170 . 171-180
Problems only . . . 101-110 . 111-120 . 121-130 . 131-140 . 141-150 . 151-160 . 161-170 . 171-180
Probs & answers . 181-190 . 191-200 . 201-210 . 211-220 . 221-230 . 231+
Problems only . . . . 181-190 . 191-200 . 201-210 . 211-220 . 221-230 . 231+
 
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