Note: The previous puzzles have been reruns of puzzles that I created for the Puzzle of the Week Contest that I run at Thompson Rivers University in Kamloops, British Columbia, Canada. Now that the academic year has started, counting this puzzle, the next ten puzzles will be new for that contest.
Triangular numbers are numbers that are the sum of the integers from 1 to some positive integer. Call the function T(n). T(1) = 1, T(2) = 1 + 2 = 3, T(3) = 1 + 2 + 3 = 6, and so on.
Some numbers can be expressed as T(a) – T(b) for some a and b. For example, 12 because T(5) – T(2) = 15 – 3 = 12.
Prove that for all integers n ≥ 2, that there are values a and b such that T(a) – T(b) = n.
Hint: There is a very simple solution which you may find if you look at the problem right. Drawing diagrams may help.
Submit your answer to Gene Wirchenko <email@example.com>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, September 25, 2013 at noon Pacific Time. I will post the answer shortly after.
There is a saying that one gets what one rewards.
This might not be what you intended to reward. People will play a system.
I saw an example of this when I was working as a shipper/receiver.
The company I was working for dealt in industrial units that were often in boxes of at least several cubic feet. Some outfits would ship units on flats. Some would use something skimpier, perhaps just 2×4s.
This made it much more awkward and unsafe to deal with the units. It was harder to get the units on the forklift, the possibility of damage to the units was much higher, and it was not as safe. Occasionally, it required two people instead of the one it would have taken with flats.
I suppose that the cost accountants looked at the expense of wood used by their warehouse and figured that they were saving money. What I do not think they saw was the additional labour since that would be buried in all of the warehouse labour. Although it appeared that going cheap on the wood saved money, I would not be surprised if, in the end, it cost more.
Hidden factors can bite.
From Kamloops’s “The Daily News”, Thursday, 2013-03-07, p. C7, The New York Times Daily Crossword:
30 Across: “Tear up”, 3 letters
The issue: “Tear” can be pronounced as rhyming with “air” or “ear”. If the former, the answer could be “rip”. If the latter, the answer could be “cry”. Per the answer the next day (Friday, 2013-03-08) on p. C5, it was the latter. Most people would probably have expected the former.
In Puzzle #13: Selling Grandma, the family was discussing the possibility of selling Grandma in order to save the old homestead. A buyer was found, and Grandma is very warm and happy at night.
After paying out the mortgage, there was some money left over. The amount was less than $50,000 and was an integer number of dollars. Divided by the first prime, the remainder is 1; divided by the second prime, the remainder is 2; and so on up to six. How much money was left over?
Submit your answer to Gene Wirchenko <firstname.lastname@example.org>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, September 18, 2013 at noon Pacific Time. I will post the answer shortly after.