This is another puzzle from BL’s Weekly Math Games.
“a + b + c = 2, and
a2 + b2 + c2 = 12
where a, b, and c are real numbers. What is the difference between the maximum and minimum possible values of c?”
The original problem statement mentioned a fourth real number d, but I considered it a typo, since it was not involved in the problem.
See Sphere and Plane Puzzle for a solution.
Presh Talwalkar has an interesting new problem.
“Students and teachers found a recent test in New Zealand to be confusing and challenging for covering topics that were not taught in class.
For the equation below, find the value of k for which the equation has numerically equal but opposite signs (for example, 2 and –2):
”
The problem didn’t mention how old the students were, but the solution to another problem on the test indicates they needed to know calculus.
See Mystery Quadratic for a solution.
This is a nice problem from Five Hundred Mathematical Challenges.
“Problem 62. A plane flies from A to B and back again with a constant engine speed. Turn-around time may be neglected. Will the travel time be more with a wind of constant speed blowing in the direction from A to B than in still air? (Does you intuition agree?)”
See Air Travel for a solution.
This is a 2017 Center of Mathematics Problem of the Week.
“The average age of a 10 member committee is the same as it was 4 years ago, because an older aged member has been replaced by a younger aged member. How much younger is the young member than the older member they replaced?”
See the Young and Old Problem.
… I know I feel differently in the morning, after seeing the pictures and reading the stories of lives arbitrarily snuffed out and people who had little reduced to people who have nothing at all. But tonight I’m going to bed in a vicious mood, with a stomach so full of contempt for this poisoned republic and its brain-dead citizens that I can taste it in my mouth, like bile.
And the only thing I can think to ask God—if she does exists—is why, just for once, can’t you smite the wicked instead of the innocent?
Posted by billmon at 01:09 AM
(Billmon’s full post can now be found on the Internet Wayback Machine at https://web.archive.org/web/20051001001007/http://billmon.org/)
This 24 September 2005 post was made during the height (or depths) of the Iraq War during the Bush-Cheney regime. Also recall the other Bush debacle, Hurricane Katrina, was just the month before, August 23–31, 2005. We thought we were in the depths of depravity and had no inkling that we were only at the higher rings of hell.
This is an interesting algebra problem from BL’s Weekly Math Games, which is behind a subscription wall.
(a3 – b3)/(a – b)3 = 73/3,
what is a – b?”
In fact, it is possible to solve for a and b individually as well.
See a Tricky Ratio Puzzle for the solution.
(Updates 11/6/2024, 11/9/2024) Other Solutions Continue reading
This is a fairly straight-forward problem from the 1999 AIME problems.
“The two squares shown share the same center O and have sides of length 1. The length of AB is 43/99 and the area of octagon ABCDEFGH is m/n where m and n are relatively prime positive integers. Find m + n.”
See the Another Octagonal Area Problem for solutions.
I have written about this a bit in my “Symbolic Algebra Timelines” post as part of the discussion of how Greek mathematics got transmitted to the present.
I had read an article by the renowned literary historian Gilbert Highet some sixty years ago which I never forgot. It discussed in some detail the miracle of the survival of records from the past. I tried to find a copy online, but in vain. So I purchased a used 1962 copy of the now extinct Horizons that contained the article and proceeded to digitize it. Since the subject of the article is about disappearing writings of the past and how some managed to survive through reproduction, I feel it somewhat apropos that I allow it to see the light of day again. He is a marvelous writer and covers the subject with fascinating detail.
Even with all the losses Highet records, writings still survived because they were in some sort of hard copy form. By 1962 fragile documents such as newspapers were being moved to microfilm. But microfilm was often replaced by magnetic tape, which was in turn replaced by a succession of digital media: floppy discs, CDs, DVDs, memory sticks, and so on. Finally, local media is being replaced by files in the “cloud”. All these media are subject to deterioration and loss, or the whims of the custodians. And of course, they all need machines and software to retrieve their information—technology which may no longer exist. Terabytes of moon data are lost on Ampex magnetic tapes for which there are no analog tape readers or even records of the data formats. Even now, some books are being published electronically and not in hard copy. Some people have referred to this ephemeral situation as the “digital Dark Age” where everything can be lost—often in an instant. So perhaps we are lucky that records from the past were not digital.
My copy of Highet’s article appears in two forms: a full, but very large, version with all the color figures, Survival of Records (58 MB), and a text-only smaller version, Survival of Records wo figs (700 KB).
Presh Talwalkar presented an interesting puzzle that originated in the Ladies’ Diary of 1739-40, was recast by Henry Dudeney in 1917, and further modified using American money.
“Each of three Dutchmen, named Hendrick, Elas, and Cornelius has a wife. The three wives have names Gurtrün, Katrün, and Anna (but not necessarily matching the husband’s names in that order). All six go to the market to buy hogs.
Each person buys as many hogs as he or she pays dollars for one. (1 hog costs $1, 2 hogs are $2 each, 3 hogs cost $3 each, etc.) In the end, each husband has spent $63 more than his wife. Hendrick buys 23 more hogs than Katrün, and Elas 11 more than Gurtrün. Now, what is the name of each man’s wife?”
See the Three Dutchmen Puzzle for solutions.
