Four identical spheres, each with diameter 20cm, are placed in a cylinder. The cylinder gives room for three of them to fit tightly at the bottom of the cylinder. The fourth sphere is placed on top of the other three. How much liquid is needed to cover the four spheres?
Problem source: Mattenøtt, NRK.
He who asks is a fool for five minutes, but he who does not ask remains a fool forever. – Chinese proverb
Eight Australian 50 cents can be placed on an Australian 10 dollars note as shown above. What is the percentage of the note not covered by the coins?
We have everything, and that’s all we have. – Ole Paus
(The original in Norwegian is even better: Vi har alt, og det er alt vi har. Even in Spanish it sounds better: Tenemos todo, y eso es todo que tenemos.)
The concept of numbers was created by many groups and civilizations independently from each other throughout the world. In the American indigenous populations we had our own mathematical systems and calculations despite never having previous contact with the Eastern Hemisphere prior to 1492 (Or? the Vikings who came before them).
One problem, many solutions.
Dilation is where the polygon grows or shrinks but keeps the same overall shape. It’s a little like zooming in or out on a camera. In the figure above, the polygon is a rectangle ABCD. As you adjust the slider on the right, the transformed rectangle A’B'C’D gets bigger and smaller, but remains the same shape.
If dilation does not disillusion you click here.
- My assistant is a clown!
- Good for you! I have a hard time finding clowns these days.
- Do you know what he did today?
- My suspicion is that I will in a minute.
- I had two equal glasses of water on a scale.
- Did the scale tip?
- No, the glasses had the same amount of water in them.
- So what is the problem?
- The clown put his fat finger in one of the glasses.
- What happened?
- I had to fire him. He ruined the experiment!
- I meant, what happened to the scale?
Whatever women do they must do twice as well as men to be thought half as good. Luckily this is not difficult. – Charlotte Whitton
Part of mathematics grows from useful problems in the real world and part from useless problems created by a playful mind. Today’s problem may well not increase the tomato harvest.
Six frogs are sitting on the vertices of a regular hexagon. Their aim is to get one of them to the center of the hexagon where a tomato is sitting. A frog can jump over any other frog and will land twice as far from the frog as it were before the jump.
Problem source: The Emissary Spring 2009.
Evaluate their ability to learn, which is far more important than what they know. – Robert L Read