Category Archives: Puzzles and Problems

Spy Gift Problem

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

Timing the Car

This is yet another simple problem from Henry Dudeney.


“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

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.

See the Distance to Flag Problem

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

Two Containers Mixing Puzzle

This is a slightly different type of a mixture problem from Dan Griller.

“Two containers A and B sit on a table, partially filled with water.  First, 40% of the water in A is poured into B, which completely fills it.  Then 75% of the water in B is poured into A, which completely fills it.  80% of the water in A is poured into B, which completely fills it.  Calculate the ratio of the capacity of container A to the capacity of container B, and the fraction of container A that was occupied by water at the start.”

See the Two Containers Mixing Puzzle

Circles in Circles

Here is another problem from the “Challenges” section of the Quantum magazine.

“Inside a circle there are two intersecting circles. One of them touches the big circle in point A, the other in point B. Prove that if segment AB meets the smaller circles at one of their common points, then the sum of their radii equals the radius of the big circle. Is the converse true?  (A. Vesyolov)”

See Circles in Circles

Two More Jugs

Here is another classic example of the three jug problem posed in the Mathigon Puzzle Calendars for 2017.

“How can I measure exactly 8 liters of water, using just one 11 liter and one 6 liter bucket?”

It is assumed you have unlimited access to water (the “third jug” of at least 17 liters).  You can only fill or empty the jugs, unless in poring from one jug to another you fill the receiving jug before emptying the poring jug.  (Hint: see the Three Jugs Problem.)

See Two More Jugs.