This is a recent Alex Bellos problem that supposedly can be solved by 12-year-olds!
“Today’s problems come from Axiom Maths, a charity that that takes high-attaining primary school children and provides them with maths enrichment during secondary school.
One of Axiom’s main activities is to organise ‘maths circles’, in which small groups of pupils get together to tackle fun problems. Such as the ones below, which are aimed at children aged 11/12, and form the basis for further explorations.
Really Secret Santa
A group of nine secret agents: 001, 002, 003, 004, 005, 006, 007, 008 and 009 have organised a Secret Santa. The instructions are coded, to keep the donors secret.
- Agent 001 gives a present to the agent who gives a present to agent 002
- Agent 002 gives a present to the agent who gives a present to agent 003
- Agent 003 gives a present to the agent who gives a present to agent 004
- and so on, until
- Agent 009 gives a present to the agent who gives a present to agent 001
Which agent will agent 007 get her present from?”
See the Spy Gift Problem
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:
See Logical Dead End
Old Codger Rant: Continue reading
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.” ”
See Timing the Car
Here is a fairly computationally challenging 1994 AIME problem .
“Find the positive integer n for which
⌊log2 1⌋ + ⌊log2 2⌋ + ⌊log2 3⌋ + … + ⌊log2 n⌋ = 1994.
where for real x, ⌊x⌋ is the greatest integer ≤ x.”
There is some fussy consideration of indices.
See the Special Log Sum.
My cousin sent me this query from the dubious Quora:
“In the Book of Genesis, only 8 humans, Noah and his sons and their four wives, survived the Flood. How many people could a family of 8 procreate in, say, 500 years?”
See Noah and Population Growth
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.
See the Amazing Root Problem
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
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.
See the Distance to Flag Problem
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
This is yet another series offered by Presh Talwalkar.
“What is the value of the following sum?
Talwalkar gives hints for three possible approaches to the solution.
See Another Challenging Sum