“Imagine an infinite chessboard that contains a positive integer in each square. If the value in each square is equal to the average of its four neighbours to the north, south, west, and east, prove the values in all the squares are equal.”—Zeitz, Paul, The Art and Craft of Problem Solving, Second Edition, p. 82 (problem 3.2.7)
Dr. Zeitz made an assumption which is not necessarily true. If you break with this assumption, you can have the value of a square be the average of its four neighbours to the north, south, west, and east, and yet the values in all the squares are not equal.
What is the assumption? Give at least one example that breaks with the assumption.
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, August 28, 2013 at noon Pacific Time. I will post the answer shortly after.
There has been considerable attention lately to how much data is collected about people. Marketers have been very busy. The NSA and other intelligence agencies have been very busy.
But no matter how much data one collects, it is always incomplete. Thinking that one has it all can lead to mistakes.
http://www.infoworld.com/t/cringely/someone-spying-your-google-searches-its-not-who-you-think-224018 by Robert X. Cringely has a very interesting section entitled “Predictive search, unforeseen consequences” which starts:
Still, all of that pales to what happened yesterday — or at least, what we thought happened yesterday. Freelance journalist Michele Catalano published a piece in Medium about a visit her husband received from members of a joint terrorism task force at their home on Wednesday. Six plainclothes officers with badges and guns pulled up in black SUVs at their home in East Meadow, N.Y., and proceeded to search their house and ask detailed questions about their Google habits.
Michele Catalano wrote in http://medium.com/something-like-falling/2e7d13e54724, “I had researched pressure cookers. My husband was looking for a backpack.” The Suffolk Police Department was concerned about pressure cooker bombs.
I wonder how many people have looked up pressure cooker bombs as a result of this story. I myself had a look.
If you are innocent, what have you got to hide? Maybe, peace of mind.
From http://www.itbusiness.ca/IT/client/en/CDN/News.asp?id=67317, paragraph 26: ‘”I think [the Galaxy S III] is going to be a massive hit for Samsung, but it isn’t as disruptive as previous devices have been because the bar is set very high now,” said Wood.’
The issue? Is it going to be a good hit (a success) or a bad hit (as in badly affecting Samsung’s bottom line)? Apparently, a good hit.
Take four red (R) beads and four green (G) beads. Arrange them in a line numbering them from 1 (left end) to 8 (right end). How many pretty arrangements are there? One of the conditions is unusual; how? Use the definitions below.
A group is all of one or more consecutive beads of the same colour. (Given RRRG, then RRR and G are groups, and RR is not a group (since it does not consist of all of the consecutive, red beads).
The size of a group is the number of beads in the group.
An arrangement of beads is pretty if it meets one or more of the following conditions:
- The pattern is symmetric. (Bead 1 is the same colour as bead 8, and similarly for 2 and 7; 3 and 6; and 4 and 5.)
- The pattern consists entirely of one or more subpatterns of a group followed by a group of the other colour and the two groups have the same size. The size of the groups in different subpatterns need not be the same. e.g. RGRRGGRG, RGRGRGRG, and GGGRRRGR are all pretty by this condition.
- The pattern consists of three groups. The size of the first group plus the size of the second group equals the size of the third group.
Submit your answer to Gene Wirchenko . Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, August 21, 2013 at noon Pacific Time. I will post the answer shortly after.