Chocolate cutting trick

https://www.youtube.com/watch?v=dmBsPgPu0Wc

I recently watched this video which showed someone cutting up a bar of chocolate. Originally I questioned where the maths was in this, until I looked further into this. I watched this video several times and tried to work out how a bar of chocolate could be cut up, with a piece taken away but still fit together looking as if the bar of chocolate never changed at all (I wasn’t complaining that bar didn’t get smaller!)

I wrote a list of instructions to work this puzzle out:

1. Remove the red rectangle
2. Move the green trapezoid to the left
3. Move the blue trapezoid to the right

This is the bar of chocolate which is not yet cut, but shows the lines where it would be cut into the different pieces. I then moved all the pieces like in the video and discovered that the car of chocolate, unfortunately does get smaller. But how does that work when it still looks the same? The picture below shows that when you take the single rectangle away, the bar of chocolate gets smaller. So how do this work? What’s the mathematics behind it?

So, where’s the maths in all of this?

First, we need to find the area of the original bar of chocolate.

Length = 6cm

Width = 4cm

Area = Length x Width

Area = 6cm x 4cm

Area = 24cm²

So, to calculate how much surface area is left when the piece is taken away we need to work out what size each piece of chocolate is and in this case, I have worked it out to be 1cm. So, finally to work this out is rather simple. All that we need to do now is find the surface area of the single piece which is 1cm x 1cm which is 1cm² and then take this away from the total surface area.

24cm²  – 1cm²  = 23cm²

So the infinite bar of chocolate, isn’t really infinite at all!

Liping Ma (2010, p.111) states that “mathematics is not rigid.”

Originally, I would have completely disagreed with this statement as I believed that mathematics wasn’t important and not needed in real life. I thought that after doing maths in school I would never use it ever again because it was pointless.

However, I discovered mathematics when baking a lemon meringue pie, which I had never thought about before.

The first piece of maths I noticed was in the recipe.

For the pastry

• 175g plain flour
• 100g cold butter, cut in small pieces
• 1 tbsp icing sugar
• 1 egg yolk

For the filling

• 2 level tbsp cornflour
• 100g golden caster sugar
• finely grated zest 2 large
• 125ml fresh lemon juice (from 2-3 lemons)
• juice 1 small
• 85g Butter, cut into pieces
• 3 egg yolk and a 1 whole egg

For the meringue

• 4 egg white, room temperature
• 200g golden caster sugar
• 2 level tsp cornflour

I never noticed before that there were so many mathematical terms used in baking such as number, kilograms, grams. When thinking about this mathematically, it is important to follow the recipe as this is key to making the best pie that I possibly could. If I had added too many eggs or butter, the pie would probably be inedible.

The next piece of maths that I discovered whilst baking was the mathematical sequencing which is more important than I had thought it would be. When it came to making the lemon curd for the center of the pie, it is important that the cornflour, sugar and lemon zest were mixed in a pan first before adding the melted butter and beaten egg yolks. This is important as if the eggs were to be added first, the recipe would not work. This is an important feature to take into consideration. This also applied to making the meringue, as I had to ensure that I had to beat the egg whites before adding in the sugar.

Other maths, I noticed in the recipe was the temperature in which I had to cook each individual section of the pie. If I had not taken the temperatures into consideration then the pie would have either been burnt or completely undercooked (and not eaten)

In conclusion, I think that baking has a lot more maths included than I had originally realised. I had to be able to understand mathematical language, be able to weigh ingredients correctly and sequence the recipe to get the best result possible! However, it didn’t quite go to plan (but it all got eaten!)

Can animals count?!

My initial reaction Richard’s question was – No, they can’t. That’s impossible!!

However, after being told about Clever Hans, the horse that could count in 1891, I thought that maybe it was possible for animals to count! Clever Hans the horse was put in a show by his owner, William Von Osten, who tried to convince people that his horse could actually count. Clever Hans would stamp his hoof off the ground the number of times that he was asked. He also claimed that Hans was able to calculate sums in his head so when given a mathematical sum such as 2 + 5, Hans would stamp his hoof 7 times. It was eventually discovered in 1904 by Oskar Pfungst that the horse was responding to subtle physical cues given by his owner.

The next animal that is said to be able to count were chimps. This time, I thought that it is likely that a chimpanzee would have the ability to count as they are the closest animal to human beings. After watching the clip about Ayumus, the chimpanzee that can count, I believed that it is possible for chimpanzees to be able to count like humans can. Ayumus was given the numbers 1-9 and they would flash up on the screen for less than 1 second but they were jumbled on the screen. Ayumus was able to put these numbers in order in a matter of seconds (if Ayumus got the numbers in the correct order, it would receive some peanuts). Ayumus managed to beat the human beings who took part in this test as they were unable to get the numbers in the correct order and could not complete the test. I believe that chimps may have the ability to compute ordinality by putting numbers in order but, I also believe that they are unable to compute cardinality (where you get to with counting is how many you have). In my opinion, I think that if the chimpanzee is getting peanuts as a reward for this test then it will almost always get the right answer!

Discussing whether or not animals can count, this got me thinking about other animals and what different species may possess this ability. So after having researched this I discovered that according to the new scientist website there are 8 animals that can count. These are:

• Salamanders – Salamanders, given a choice between tubes containing two fruit flies or three, lunge at the tube of three1. This hints that the ability to differentiate between small numbers of objects may have evolved much earlier than scientists had thought.
• Chicks – Experiments show that new born chicks have an innate sense of number. Chicks always try to stay close to objects they are reared with – just as they stay close to and follow their mother as soon as they hatch. This instant recognition is known as “imprinting”. When chicks are separated with their mother, they will always go towards a large number of things that look like themselves or their mother.
• Horses – Researchers found that, when offered a choice, they consistently choose buckets containing higher numbers of apples. Babies aged from 10-months-old have been shown to have an innate tendency to go for containers holding larger numbers of food items. Thus showing that horses have the ability to recognise quantities.
• American Coots – Studies show that coots can trick others into raising their eggs for them. When coots lay their eggs, they can count them to see how many that they have laid. Other hens will try to sneak their eggs into another’s nest but the coot will reject the rogue eggs that they have not laid themselves.
• Monkeys – Studies have shown that monkeys, like horses have the ability to understand quantity. When given the task to pick out the group of dots which is the largest, the monkey was able to do so (there was probably a reward for this).
• Mosquitofish – It was known that fish could tell big shoals from small ones, but researchers have now found that they have a limited ability to count how many other fish are nearby. This means that they have similar counting abilities to those observed in apes, monkeys and dolphins and humans with very limited mathematical ability.
• Lemurs – Again, like monkeys and horses, lemurs have the ability to determine quantity. They are able to pick the larger quantity of food!
• Honeybees – Tests showed that honeybees were able to tell the difference between patterns containing two and three dots, and researchers believe they could be trained to recognise four dot patterns as well.

Overall, I think that animals are a lot more intelligent than I originally thought. If they can count, I’m still unsure.