Solutions to Math Puzzles
The Monthly Question: December 2008
The Monthly Question: December 2008
The question was:
I shared a warehouse of oranges with my alphabetic friends. I gave half the oranges plus half an orange to Mr A. Then, I gave half the remaining oranges plus half an orange to Mr. B. I kept doing this for a representative of every letter of the alphabet, until lastly, Mr. Z received half the remaining oranges plus half an orange. No fruit was left over. How many were in the warehouse?
Solution
The number of oranges is given by 2n-1, where n = number of people. So 226-1= 67 108 863 oranges. How do you arrive at such a formula?
Well, Mr A received (x+1)/2 oranges. Mr B received (x+1)/4. Assuming a simpler case in which there are only two recipients, then the sum of these expressions would equal x. Solving for x we would get 3. If there was a third recipient, Mr C would get (x+1)/8. Solving would yield 7. Mr D would lead to an x value of 15, in each case 1 less than a power of 2 corresponding to the number of recipients.
The Monthly Question: November 2008
The Monthly Question: November 2008
The question was:
How did President Garfield use the following to prove the Pythagorean theorem?
Solution
The area of the trapezoid can be expressed as (a+b)(a+b)/2 = ab/2+ab/2+c2.
(a2 + 2ab + b2)/2 = ab/2 + ab/2 + c2.
or a2+b2=c2.
The Monthly Question: October 2008
The Monthly Question: October 2008
The question was:
Without any graphing or algebraic manipulations, find the values of x and y:
2x + 5y = 8.
9x +12y =15.
Solution
Whenever the coefficients of each equation go up by a constant interval, the solution is (-1,2)
Here's why:
For a system of equations
Ax + By= C
Dx + Ey = F
Ax + (A+q)y = A+2q
Dx + (D +r)y = D+2r
D[Ax + (A+q)y = (A+2q)]
-A[Dx + (D +r)y = (D+2r) ]
Distribute and add the preceding:
DAx + ADy + Dqy = AD + 2Dq
-DAx –Ady –Ary = -AD -2Ar
Dqy –Ary = ACD + 2Dq - AD -2Ar
Simplify and factor the above : y(Dq-Ar) = AD+ 2Dq - AD -2Ar
y(Dq-Ar) = 2(Dq – Ar)
y = 2
Substitute y=2 into one of the originals and x = -1
Monthly Question: September 2008
The question was:
According to a bus schedule, there should be a bus every 15 minutes. What is the average waiting time?
Solution
Depending on when you arrive at the stop, your waiting time can be anywhere from 0 to 15 minutes ( if you've just missed it. ) Those are 16 different possibilities for a sum of 15(16)/2 minutes. Dividing the latter by the former you get an average waiting time of 7.5 minutes.
Monthly Question: August 2008
The question was:
An Italian couple wants to pack their trunks with 140 lbs of imported pasta, which they think will be expensive ( $1.50 to $2 American per pound) in their new destination. They buy the pasta at 30 cents Canadian per pound, but have to pay $3.45 Canadian per kg to fly it over. Should they bother or is it cheaper to buy it in the U.S. ?
Solution
To buy it in Canada and bring it to the U.S. it will cost (0.30 + 3.45/2.2)= $1.87 Canadian per pound. (The second rate has to be converted from kg to lb.) If they buy it in the U.S., with a currency exchange of 1 U.S. = $1.40 Canadian, it will cost between $2.10 and $2.80 Canadian per lb. So if the assumed prices hold true,and if there are no customs duties, then it is worth buying in Canada.
Monthly Question: July 2008
The question was:
A certain six digit number is of the form abcabc. One of its factors is 979. Find the smallest number with such characteristics.
Solution
If the number is of the form abcabc, then the number abc is a multiple of 1001. Let the three digit number abc = x. 1001 x = 979 y . The greatest common factor of 1001 and 979 is 11. Dividing through we obtain: 91 x = 89 y . Since the g.c.f. of 91 and 89 is 1, y must be a multiple of 91. Since x is a three digit number, the smallest possible value for y = 182. x = 89*182/91 = 178. So the smallest six-digit number of the form abcabc and which is divisible by 979 is 178178.
Monthly Question: June 2008
The question was:
Consider the following common trick:
1) Pick a number from 1-9.
2) Subtract 5.
3) Multiply by 3.
4) Square the number.
5) Add the digits until you get only one digit (i.e. 64 = 6+4= 10 = 1+0 =1)
6) If the number is less than 5, add five. Otherwise subtract 4.
7) Multiply by 2.
8) Subtract 6.
9) Map the digit to a letter in the alphabet 1= A, 2= B, 3 =C, 4 = D etc...
10) Pick a name of a country that begins with that letter.
11) Take the second letter in the country name and think of a mammal that begins with that letter.
12) Think of the color of that mammal.
Why does it work? How do I know you'll end up with a grey elephant from Denmark ?
Solution
Let x be the original number.
After step 4, you have 32( w - 5)2.
Since you were limited to picking a number from 1 to 9, by substituting these into the above expression, we see that your answer so far will either be 9, 36 , 81 or 144*, each of which will yield an answer of 9 after the fifth step.
So, no matter what your original number was, you will end up with 9 after step 5 and the letter D after step 9.
There are five countries that begin with the letter D. But after a minute of mental arithmetic, most people will not think of Djibouti, Dominica, Dominican Republic or Dutch Guiana.
Similarly, for a mammal, most people will not immediately think of echidna, edentate, eland, elk, ermine, elkhand or of an elephant seal for that matter.
* More generally, let x, y and z... be the digits of the result of 32( w - 5)2.
Then 32( w - 5)2 = z + 10y + 100x
But we are just adding the digits as if they were ones ; so imagine that 99x and 9y have been subtracted from both sides:
x + y + z = 32( w - 5)2 - 99x - 9y
or x + y + z = 9[ ( w - 5)2 - 11x - 1y ]
so clearly the sum of the digits is a multiple of nine.
For example, if w = 101,
32(101-5)2 = 82944
