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Golden Thoughts

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?


A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?

Nutrition and Cycling

Age 14 to 16
Challenge Level
Lots of people attempted the first part of the question - thanks to all those who submitted solutions! Anna, Millie, Deena, Lizzie and Izzy, from The Castle School Academy in Taunton, had some of the following ideas:

The ride is 38 miles long, which is 38 x 1.61 = 61.18 km. Multiplying by the 100 kJ of enengy used to cycle each kilometer, Andy will use 6118 kJ by cycling. This is converted to kcal using the fact that 1 kcal = 4.19 kJ, so 6188 kJ = 6188 / 4.19 = 1460 kcal.
Since Andy had a big meal of 800 kcal for breakfast he has 1460 - 800 = 660 kcal to make up on the journey.

Even though the ride is short, Anna, Millie, Deena, Lizzie and Izzy wanted to account for the 2500 kcal that Andy needs on top of his exercise requirements. They assumed that Andy burns these 2500 calories at a constant rate during the 24 hour day.

On top of the exercise, during the 2 hour ride Andy needs 2 x 2500 / 24 = 208.3 kcal anyway, which means a total of 1460 + 208.3 = 1668.5 kcal need to be consumed. He has a big meal (800 kcal) for breakfast, so he needs to make up 868.5 kcal on the ride. If he eats 7 bananas, he can make up 840 kcal; if he eats 8 cereal bars, he can make up 800 kcal.

Good! But, didn't Andy only want to limit himself to 250kcal per hour on short trips? What happens here? Jim, from Strathallen, has the answer:

If he wishes to eat bananas, he can eat four of them and consume 480 kcal. If he wants to eat cheap cereal bars, he can eat five of them and consume 500 kcal. So the cereal bars minimises his defecit.

Great - thank you!

Ewan, from King Edward VII School in Sheffield, tackled the second problem with the following method: 

The distance from Land's End to John o' Groats is 1407 km, which is 1407 / 1.61 = 874 miles. We know that Andy cycles at 14 mph on multi-day trips so the cycling will take him 874 / 14 = 62.4 hours. Since he wishes to cycle for no more than 7 hours per day, and 62.4 / 7  = 8.9, Andy will take 9 days to complete the trip.

Ewan assumed that Andy would cycle for 7 hours on the first 8 days and have a slightly shorter last day, then worked out the calorie defecit.

In a 7 hour day Andy travels 7 x 14 = 98 miles, which is 98 x 1.61 = 157.78 km. This needs 15778 kJ of energy which is 3766 kcal. Adding the 2500 kcal needed in addition to these calories, Andy requires 6266 kcal per day.
The three really big meals provide 3000 kcal, leaving 3266 kcal from snacks. He can have 350 kcal per hour for 7 hours in on-the-road snacks which is 2450 kcal. This leaves 816 kcal to get from off-the-road snacks.

Another student, Emily, worked out several ways of achieving 2450 kcal (or more!) from snacks and energy drinks:

Andy can carry 3l of energy drink which provides a total of 6 x 190 = 1140 kcal (and he can then refill with water to provide his fluid needs for the rest of the ride).

He can carry
8 bananas = 960 kcal
or 8 energy bars = 1760 kcal
or 16 cheap cereal bars = 1600 kcal
or 24 energy gels = 2640 kcal (though he wouldn't want to consume more than 2450kcal worth)

So he can meet his energy needs with any of these supplemented by energy drink, except bananas.

You might like to think of letting Andy carry combinations of different snacks, he does have 8 pockets after all. See if you can find a way of getting exactly 2450 kcal so that Andy doesn't carry spare food.