This is a clever puzzle from the 1986 AIME problems.
“The pages of a book are numbered 1 through n. When the page numbers of the book were added, one of the page numbers was mistakenly added twice, resulting in an incorrect sum of 1986. What was the number of the page that was added twice?”
See the Additional Page Problem for the solution.
This is an intriguing puzzle from Futility Closet.
“In Robert Chambers’ 1906 novel The Tracer of Lost Persons, Mr. Keen copies the figure below from a mysterious photograph. He is trying to help Captain Harren find a young woman with whom he has become obsessed.
‘It’s the strangest cipher I ever encountered,’ he says at length. ‘The strangest I ever heard of. I have seen hundreds of ciphers—hundreds—secret codes of the State Department, secret military codes, elaborate Oriental ciphers, symbols used in commercial transactions, symbols used by criminals and every species of malefactor. And every one of them can be solved with time and patience and a little knowledge of the subject. But this … this is too simple.’
The message reveals the name of the young woman whom Captain Harren has been seeking. What is it?”
As is usual with these types of puzzles, I felt foolish that I couldn’t see the immediate, simple interpretation of the boxes—after a great deal of effort. So I solved it using the usual cryptographic methods that rely heavily on logic and letter frequencies, though the message is a bit short for that.
See Box Code Puzzle for solutions.
In one of our periodic FaceTime calls I found out that my granddaughter in 6th grade was interested in learning algebra and had gotten a book to help her out. Clearly this initiative to get a head start prior to the normal course curriculum excited me, so I wrote what I thought was an insightful essay on the meaning and purpose of algebra. Needless to say it was an abysmal failure.
That got me to thinking deeply about what it meant to learn mathematics and in particular symbolic algebra.
