May 2009

Mathematical puzzles for your intellectual amusement.

April 29 Problem. In honor of Ron Milne’s retirement, the following problem involving bicycles and geometry is offered. It turns out that bicycles with square wheels can work fine – as long as the road is appropriately constructed. Can a suitable road be constructed for a bicycle with equilateral triangle wheels? If so, how? If not, why not?

Solution by Nick Bouwman. Yes, as long as the arc length of the inverted catenary equals that of one of the sides of the triangle and the inverted catenary is cut off where it makes a 30 degree angle so that when it is joined with another inverted catenary, it forms a 60 degree angle: the size of each of the triangle’s angles.

May 6 Auction and Problem. Time for a somewhat strange auction. Send dhousman// @@goshen.edu a bid (a real number no smaller than 1) by Tuesday, May 12. The person with the highest bid pays nothing and wins the cash prize. If there is a tie, the highest bidders share the cash prize equally. The cash prize is $100 divided by the sum of the bids rounded up to the nearest penny. For example, if I were to receive bids of 1, 1.5, and 2.3, then the person who submitted the bid of 2.3 would receive $100 / (1 + 1.5 + 2.3) = $20.84. Although unwise from your perspective, multiple bids will be accepted. All you need to do is bid, but some may wish to explain the reasons for their bids. One of the usual assortment of non-cash prizes (T-shirt, chocolate, or toy) will go to the person with the best explanation (judged in a highly subjective manner).

Auction Winner. Seven people submitted bids.  The bids summed to roughly 38.503 (one bid was a many digit approximation of pi).  So, the winner of $2.60, with a high bid of 13.52, is Sandy Slabaugh.  Of course, if the seven bidders had made an agreement to submit only one bid of 1, then that bidder could have split $100 among the seven, netting $14.28 or so for each.

Best  Reason by  Dan Stutzman. My bid: 1.  May all bidders be winners. I chose the option that is best for the ‘greater good’ in hopes that all others thought the same. It’s a strange mix of altruism and selfishness. What is more strange: if others also took that leap and the greater good benefits me, I feel my faith in humanity is somewhat restored. But it usually isn’t. Perhaps this means that I shouldn’t mix politics and math.  Oh well, I’ll take my chances.

Best Mathematical Reason by Isaac Witmer. Assuming a perfect distribution of bids, with an average number of 5 bids (the upper bound of my estimate of participants per week), I will choose a number high enough that I can win without choosing too high that I basically win nothing.  So first we’ll assume that I won, and calculate my winnings to find a total that is an acceptable amount of money to win.  Since the total is 100, I could find about 5 dollars of that pot to be worth winning. Anything else isn’t really worth it.  So, I hope the total (assuming I’m the upper bound) to be 20. If 5 people add up to 20, with an even distribution:
1+(1+y)+(1+2y)+(1+3y)+(1+4y)=20
5+10y=20
15=10y, y=1.5
So each person will increment by ~1.5, and I should guess around 7.  My Bid is 7.

Best Short Reason by Emily Shantz. hmm, i was going to ask if i could bid infinity, but then i realized that that would take the cash prize down to like, nothing. so my official bid is: 5.7.

Best Evangelizing Reason by Nina Mishler. My bid is 3.14159 or pi.  My reason for bidding is this. I have been influenced by the Hoosier Lottery motto “You have to play to win.”  The only time I buy lottery tickets is to tie on the tops of Christmas and birthday packages.  Probably also unwise from my perspective.   I had to think if I knew what a real number was.  I couldn’t come up with anything so I went to Wikipedia.  It was interesting to read about real numbers.  It renewed my love for the mysteriousness of math.  My eye was drawn to the word pi.  I read a book once titled the Life of Pi (not about numbers at all).  So I chose this number for it’s mysteriousness and hoping it would be the highest number of the lowest value to win the most amount of money or even some chocolate.  I think it’s important that the admission office participates in the math problem of the week because we are constantly asked about “the numbers.”  I am sending this math puzzle to my admitted through deposited students considering GC for Fall 2009 that are math or math education majors.

May 13 Problem. PEACE + BY + PEACE = GOSHEN. Make the statement arithmetically true by assigning a different digit to each different letter (except that B and Y share the same digit).

Solution by Greg Yoder. 69209 + 77 + 69209 = 138495.