Logical Dead End

One is reduced to hysterical laughter to try to maintain a modicum of sanity.

Our Senate at work: Republican Mitch McConnell said (Dec 6) “Legislation that doesn’t include policy changes to secure our borders will not pass the Senate.”  Republican Trump said (Feb 3) the Senate should not pass legislation that includes border security.  Let P be the statement “Senate legislation should include border security.” and let Q be the statement “Senate should pass legislation.”  Then we have the Republicans saying

(~P ⇒ ~Q) ˄ (P ⇒ ~Q)

Show that this is equivalent to ~Q, that is, “The Senate should not pass legislation.”—basically stop working.

It looks like the Republicans in the House are doing the same thing:

politico.com

See Logical Dead End

Old Codger Rant, with Update (4/24/2024):  Continue reading →

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Timing the Car

This is yet another simple problem from Henry Dudeney.

“57. TIMING THE CAR

“I was walking along the road at three and a half miles an hour,” said Mr. Pipkins, “when the car dashed past me and only missed me by a few inches.”

“Do you know at what speed it was going?” asked his friend.

“Well, from the moment it passed me to its disappearance round a corner I took twenty-seven steps and walking on reached that corner with one hundred and thirty-five steps more.”

“Then, assuming that you walked, and the car ran, each at a uniform rate, we can easily work out the speed.” ”

Answer.

See Timing the Car for a solution.

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Amazing Root Problem

Well, I discovered that the 2024 Math Calendar has some interesting problems, so I guess things will limp along for a while.  This is a challenging but imaginative problem from the calendar.

_______________

As before, recall that all the answers are integer days of the month.

Answer

See the Amazing Root Problem for a solution.

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Five Year Anniversary

So I managed to make it five years.  Again, I thought I would present the statistical pattern of interaction with the website in the absence of any explicit feedback.

But as the summary shows, the fall-off of visitors that began in the middle of last year has persisted throughout 2023.  I have also run out of much new material, so I am basically going to wrap it up.  I have a few things in the hopper, but they are mostly similar to puzzles already presented.  I have one or two essay ideas left, but again I have mostly said what I have to say, and the world of math has moved on.

Anyway, here is the summary for what it’s worth.

See Five Year Anniversary

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Distance to Flag Problem

The following puzzle is from the Irishman Owen O’Shea.

“The figure shows the location of three flags [at A, B, and C] in one of the fields on a neighbor’s farm.  The angle ABC is a right angle.  Flag A is 40 yards from Flag B.  Flag B is 120 yards from flag C.  Thus, if one was to walk from A to B and then on to C, one would walk a total of 160 yards.

Now there is a point, marked by flag D, [directly] to the left of flag A.  Curiously, if one were to walk from flag A to flag D and then diagonally across to flag C, one would walk a total distance of 160 yards.

The question for our puzzlers is this: how far is it from flag D to flag A?”

This problem has a simple solution.  But it also suggests a more advanced alternative approach.

Answer.

See the Distance to Flag Problem for a solution.

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More Right Triangle Magic

James Tanton asked to prove the following surprising property of a right triangle and its circumscribed and inscribed circles.

“Every triangle is circumscribed by some circle of diameter D, say, and circumscribes another circle of smaller diameter d. For a right triangle, d + D equals the sum of two side lengths of the triangle. Why?”

See More Right Triangle Magic

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