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Blog Support for Growing Mathematicians

Coronavirus vs SARS Graphical Error

January27

New of the fast spreading coronavirus is cause for deep concern, as well as action to contain those who have been affected. However, H3 noticed a rather erroneous graph which attempted to compare coronavirus with the SARS virus. What is wrong with this representation? (see the Post Support column for a possible answer):

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Mathematics, with a Shot of Espresso

January23

“I don’t need math ’cause I’m just going to work as a barista!” Don’t be too hasty in making a statement like this until you read this article:

“An espresso is one of the most popular ways to drink coffee, but also the most complicated to make.The process depends on a delicate combination of factors — including grind setting, coffee amount, water pressure, temperature and beverage volume — and it’s hard to consistently pull the perfect shot.

A group of researchers say they’ve found the key to an espresso that’s cheaper, consistently tasty and better for the planet, too. The secret, according to an article in the journal Matter, is to go against traditional espresso wisdom and use fewer coffee beans that are ground more coarsely.

“American companies are catching on to this idea that being able to produce a shot quickly is a positive thing, and being able to produce it reproducibly, of course, is the basis of American consumerisms,” said Christopher Hendon, co-senior author of the report and computational chemist at the University of Oregon. Though a lot of factors are involved, the norm for brewing an espresso is to grind a large amount of coffee beans — around 20 grams or 0.7 ounces — as finely as possible.
But when researchers tested these recommendations, they found the relationship between them was more complicated: grinding as finely as the industry standard clogged the coffee bed, resulting in areduced extraction yield, wasted coffee grounds and inconsistent flavors.

To develop a recipe for consistent extraction yield based on the factors under a barista’s control — the amount ofwater and coffee, size of the grounds, and water pressure — the researchers created a mathematical model that helped them narrow it down to two elements: the size of the grind and the amount of coffee used.
One way to optimize extraction and achieve reproducibility is to grind coarser and use a little less water, while another is to simply reduce the mass of coffee,” Hendon said.

Their math model was tested at a local coffeeshop in Oregon using an espresso machine that allowed for control of shot time, water pressure and temperature. By reducing the mass of coffee used by 25%, the cafe was able to save 13 cents per drink, amounting to $3,620 per year. In addition to the savings, the shot times were reduced to 14 seconds, significantly reducing the order-to-delivery time.” [source: CNN]
Now, out you like a shot of coffee with your math today?

Please try this at home or school

January20


Take a humble door knob and shine a strong flashlight to reveal a geometric shape. What is it?

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It’s a small world after all

January20

Size matters to mathematicians, as well as scientists (another name for mathematicians). Last year, scientists (aka mathematicians) at York University recorded the radius of a proton at 0.833 femtometres, or just under one trillionth of a millimetre. It’s enough to make your head (like your protons) spin!
Incidentally, and totally unrelated, H3 visited the Music Department at York back in the late 1990’s while investigating computer-based methodologies for teaching music.

Easy Speed Multiplication

January17

How Big is a Trillion?

January15


Wow, I just read in the news today that the world has amassed a debt level of just over 250 trillion dollars!

But, just how big is a trillion? A trillion is a million million, or 1,000,000 with 6 more zeros, or 1,000,000,000,000. This video puts that huge amount into perspective (click on the picture below to watch):

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How can someone born in 2020 be older than someone born in 2019?

January4

Solution: If someone is born, say, at 1am in the morning of January 1 2020 in Sydney, then everyone born in the UK in the afternoon of December 31 2019 will be younger than them but born the previous year.

In fact, the number of people born in 2020 who are older than someone born in 2019 is going to be in six figures. About 360,000 people are born every day. (Most in Asia). A child born in French Polynesia between 10pm-midnight on Dec 31 2019 will be younger than all the Jan 1 2020 births until that moment, i.e until 6pm in Beijing, 3.30pm in Delhi, and so on. A fair estimate is probably about half of 360,000.

(Hat tip to the reader who said that anyone born in 2020BC will be older than anyone born in 2019AD)

source and more puzzles such as the one below are from The Guardian

Second Puzzle: Fill in the blanks in the following ‘countdown’ equation so it makes arithmetical sense:

10 9 8 7 6 5 4 3 2 1 = 2020

[One answer in Post Support]

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All I wanted for Christmas was a Tesseract!

December27

Chocolates, clothes and a book to read are great Christmas presents, but what I really wanted was a 3D Tesseract!
“In geometry, the tesseract is the4D analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells. The tesseract is one of the six convex regular 4-polytopes.

The tesseract is also called an eight-cell, C8, (regular) octachoron, octahedroidcubic prism, and tetracubeIt is the four-dimensional hypercube, or 4-cubeas a part of the dimensional family of hypercubesor measure polytopesCoxeter labels it the polytope.

According to the Oxford English Dictionary, the word tesseractwas coined and first used in 1888 by Charles Howard Hinton in his book A New Era of Thought, from the Greekτέσσερεις ἀκτίνες(téssereis aktines, “four rays”), referring to the four lines from each vertex to other vertices.[5]In this publication, as well as some of Hinton’s later work, the word was occasionally spelled “tessaract”.” (source: Wikipedia)

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The 12 Days of Christmas – How Many Gifts?

December25

When I accept all the gifts in the song “The 12 Days of Christmas” how many gifts do I receive? And, can I work out the total using some neat Mathematics? First, a reminder about the song:

“On the first day of Christmas
my true love sent to me:
A Partridge in a Pear Tree

On the second day of Christmas
my true love sent to me:
Two Turtle Doves
and a Partridge in a Pear Tree

On the third day of Christmas
my true love sent to me:
Three French Hens
Two Turtle Doves
and a Partridge in a Pear Tree

On the fourth day of Christmas
my true love sent to me:
Four Calling Birds
Three French Hens
Two Turtle Doves
and a Partridge in a Pear Tree

On the fifth day of Christmas
my true love sent to me:
Five Golden Rings
Four Calling Birds
Three French Hens
Two Turtle Doves
and a Partridge in a Pear Tree

On the sixth day of Christmas
my true love sent to me:
Six Geese a Laying
Five Golden Rings
Four Calling Birds
Three French Hens
Two Turtle Doves
and a Partridge in a Pear Tree

On the seventh day of Christmas
my true love sent to me:
Seven Swans a Swimming
Six Geese a Laying
Five Golden Rings
Four Calling Birds
Three French Hens
Two Turtle Doves
and a Partridge in a Pear Tree

On the eighth day of Christmas
my true love sent to me:
Eight Maids a Milking
Seven Swans a Swimming
Six Geese a Laying
Five Golden Rings
Four Calling Birds
Three French Hens
Two Turtle Doves
and a Partridge in a Pear Tree

On the ninth day of Christmas
my true love sent to me:
Nine Ladies Dancing
Eight Maids a Milking
Seven Swans a Swimming
Six Geese a Laying
Five Golden Rings
Four Calling Birds
Three French Hens
Two Turtle Doves
and a Partridge in a Pear Tree

On the tenth day of Christmas
my true love sent to me:
Ten Lords a Leaping
Nine Ladies Dancing
Eight Maids a Milking
Seven Swans a Swimming
Six Geese a Laying
Five Golden Rings
Four Calling Birds
Three French Hens
Two Turtle Doves
and a Partridge in a Pear Tree

On the eleventh day of Christmas
my true love sent to me:
Eleven Pipers Piping
Ten Lords a Leaping
Nine Ladies Dancing
Eight Maids a Milking
Seven Swans a Swimming
Six Geese a Laying
Five Golden Rings
Four Calling Birds
Three French Hens
Two Turtle Doves
and a Partridge in a Pear Tree

On the twelfth day of Christmas
my true love sent to me:
12 Drummers Drumming
Eleven Pipers Piping
Ten Lords a Leaping
Nine Ladies Dancing
Eight Maids a Milking
Seven Swans a Swimming
Six Geese a Laying
Five Golden Rings
Four Calling Birds
Three French Hens
Two Turtle Doves
and a Partridge in a Pear Tree”

Here is the solution …

Partridges: 1 × 12 = 12
Doves: 2 × 11 = 22
Hens 3 × 10 = 30
Calling birds: 4 × 9 = 36
Golden rings: 5 × 8 = 40
Geese: 6 × 7 = 42
Swans: 7 × 6 = 42
Maids: 8 × 5 = 40
Ladies: 9 × 4 = 36
Lords: 10 × 3 = 30
Pipers: 11 × 2 = 22
Drummers: 12 × 1 = 12

Total = 364

And, here is another way …
The number of presents each day is 1 on the 1st, then 3 on the 2nd, then 6 on the 3rd, then 10 on the 4th. We call this set of numbers the triangular numbers, because they can be drawn in a dot pattern that forms triangles:

Another way of writing this is:

On the first day, 1 present.
On the 2nd day, 1 + 3 = 4 presents
On the 3rd day, 1 + 3 + 6 = 10 presents
On the 4th day, 1 + 3 + 6 + 10 = 20 presents.

These partial sums are called tetrahedral numbers, because they can be drawn as 3-dimensional triangular pyramids (tetrahedrons) like this:

So how many dots (representing presents) will there be in the 12th tetrahedron?

Of course, we could just start adding with our calculator, but what if my true love is very generous, and starts giving me presents for 30 days after Christmas? Or for 100 days? How would I calculate it then?

Our aim is to produce a formula that will allow us to find any tetrahedral number. Here’s one of the possible ways of doing this.

In general, for the sum 1 + 2 + 3 + … + n:

\text{Sum}={\frac{{n}}{{2}}}{\left[{2}+{\left({n}-{1}\right)}\right]}

which is the same as

\text{Sum}={\frac{{n}}{{2}}}{\left({n}+{1}\right)}

Multiplying by the (n+ 2) that we get from what I called ‘the result triangle’ earlier:

{\frac{{n}}{{2}}}{\left({n}+{1}\right)}{\left({n}+{2}\right)}

Dividing this by 3 (since we used 3 equivalent sum triangles to get this far) gives us the n-th tetrahedral number:

{\frac{{n}}{{6}}}{\left({n}+{1}\right)}{\left({n}+{2}\right)}

On the 12th day, the number of presents will be

{\frac{{12}}{{6}}}{\left({13}\right)}{\left({14}\right)}={364}

Phew! See the complete working here and, above all, enjoy a wonderful Christmas!

Pay Attention to Patterns and Templates

December21

Lessons from the life of a great mathematician – John von Neumann

« Older Entries

Post Support

The graph on the left (Coronavirus) is for a time period of 30 days, while the one on the right (SARS) is for 8 months! Very poor graphical comparison and hardly relevant, unless it is attempting to downplay the seriousness of the coronavirus?

10 x 9 x 8 + (7 + 6) x 5 x 4 x (3 + 2) x 1 = 2020

NCEA Level 2 Algebra Problem. Using the information given, the shaded area = 9, that is:
y(y-8) = 9 –> y.y – 8y – 9 =0
–> (y-9)(y+1) = 0, therefore y = 9 (can’t have a distance of – 1 for the other solution for y)
Using the top and bottom of the rectangle,
x = (y-8)(y+2) = (9-8)(9+2) = 11
but, the left side = (x-4) = 11-4 = 7, but rhs = y+? = 9+?, which is greater than the value of the opp. side??
[I think that the left had side was a mistake and should have read (x+4)?]

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