Hope and Change

Just stumbled across this in an 1889 newspaper:

To those who love mathematics, here is a simple problem for you to figure out: A man purchased groceries to the amount of 34 cents. When he came to pay for the goods he found that he had only a $1 bill, a 3-cent piece and a 2-cent piece. The grocer, on his side, had only a 50-cent piece and a quarter. They appealed to a bystander for change, but he, although willing to oblige them, had only two dimes, a 5-cent piece, a 2-cent piece and a 1-cent piece. After some perplexity, however, change was made to the satisfaction of everyone concerned. What was the simplest way of accomplishing this?

($1 is worth 100 cents, a quarter 25 cents, and a dime 10 cents.)

Click for Answer



I think Jacques Jouet was the first to notice this. In the first few pages of the Tintin adventure The Secret of the Unicorn, as Tintin visits the Vossenplein antique market in Brussels, Snowy the dog keeps scratching himself. Why?

Click for Answer

The Last Digit

A problem from the 1996 Georg Mohr mathematics competition in Denmark:

n is a positive integer. The next-to-last digit in the decimal expression of n2 is 7. What’s the last digit?

Click for Answer

RSS Quiz

Another holiday challenge: The Royal Statistical Society’s 2017 Christmas quiz presents 13 problems that require general knowledge, logic, and lateral thinking but no particular math skills. For example:

4. CAN YOU DIG IT? [11 points]

Identify the following from the clues. What do all ten answers have in common?

  • A cockerel of human dimensions, performed by a prolific informer
  • William’s book (1956), Charles’s film (1957), or Mike & Al’s song (1971)
  • Challenger first appeared here, over a century ago
  • A cricketer, a rugby player, or a commentator
  • 2001 boy-band album – a remix of “NOW FOR LOUD ROW”
  • The official title of Guinness
  • One who reigned for almost 999 million seconds
  • King’s collection, ordered by cards
  • The Bassett country residence, according to Plum
  • ATR co-founder, name-checked by “The Tiger” between The Slits and Dickens

You can use any tools or resources you like, including books, search engines, and computer programs. Anyone can enter, and you stand to win £150 if you’re an RSS member. The deadline for entries is January 7. The full quiz is here.

(Thanks, Dave.)

A Thoughtful Gift


University College London mathematician Matthew Scroggs has created a mathematical Christmas card for Chalkdust magazine.

Solve 10 puzzles, convert the answers to base 3, write them in the grid, and color them accordingly to reveal a Christmassy picture.

He’s provided both a PDF and an online version that will color the squares for you.

Two Puzzles


In 2005 Keele University computer scientist Gordon Rugg published two ciphers to the web.

The first is called the Penitentia Manuscript. The image above is only one panel; you can view and download the whole thing here. Rugg’s website provides one clue: “Most modern codes are based on a shared set of underlying assumptions. He wondered what would happen if you deliberately ignored those assumptions. What sorts of code might that produce?” There’s some more info here.

The second cipher, called the Ricardus Manuscript, was inspired by Rugg’s work on another famous puzzle: “When Gordon was working on the Voynich Manuscript, he started wondering what a real code based on the components of the Voynich Manuscript would look like. This code is the result.” Again, the image below is only a sample; you can find the whole thing here. More info here.

Both of these ciphers have been freely available on the web for more than 10 years, and both remain unsolved. Any takers?