### Consecutive Numbers

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

### Calendar Capers

Choose any three by three square of dates on a calendar page...

A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

# River Crossing

##### Age 11 to 18Challenge Level

You may have met this introductory problem before, but have a think about it in preparation for tackling the main one that follows.

### The fox, the hen and the corn

Once upon a time a farmer went to market and bought a fox, a hen and a sack of corn.

The farmer came to a river which needed to be crossed by boat. He could take only one of his purchases - the fox, or the hen, or the sack of corn - in the boat at a time.

If left together on the riverbank without the farmer, the fox would eat the hen, or the hen would eat the corn.

Can you find a way for the farmer to get all of them across the river safely? (Note - only the farmer can row the boat!)

Can you find more than one way of getting them across with the minimum number of crossings?

There are two possible ways of getting them all across in seven crossings:

Hen & Farmer $\rightarrow$
$\leftarrow$ Farmer
Fox & Farmer $\rightarrow$
$\leftarrow$ Hen & Farmer
Corn & Farmer $\rightarrow$
$\leftarrow$ Farmer
Hen & Farmer $\rightarrow$

Hen & Farmer $\rightarrow$
$\leftarrow$ Farmer
Corn & Farmer $\rightarrow$
$\leftarrow$ Hen & Farmer
Fox & Farmer $\rightarrow$
$\leftarrow$ Farmer
Hen & Farmer $\rightarrow$

Two adults and two children need to cross a river in a rowing boat. Can you determine how to get everyone across, and how many crossings are needed, given the following information?

• All adults weigh the same
• Each child weighs half as much as an adult
• The boat can only carry the weight of one adult
• Adults and children can row the boat
• The boat must have someone in it to row it!

What if there were $3$ adults and $2$ children?
What if there were $100$ adults and $2$ children?
What if there were $n$ adults and $2$ children?

If you enjoyed these problems, you may also like to take a look at Crossing the Bridge.

With thanks to Don Steward, whose ideas formed the basis of this problem.

Notes and Background

When you try to solve these 'river crossing' puzzles, you are attempting some of the same problems that were set by Alcuin, an ecclesiastic from the 9th century, for the Emperor Charlemagne.

Alcuin's version of Problem 1 concerned a wolf, a goat and a cabbage.