Since I always fail my New Year's resolutions, this year, my New Year's resolutions are:
1. Make a new year's resolution
2. Fail my new year's resolution
Am I guaranteed to succeed? Guaranteed to fail? Something else?
Let weird(n) be the number of ways to write n as a + a + b + c, where n, a, b, and c are distinct positive integers. Is weird(n) an increasing function?
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A set of positive integers has sum 25. What is the biggest you can make the product of the numbers?
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Can you work out the area of the inner square?
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The pattern 123451234512345... is continued to form a 2000-digit number. What is the sum of all 2000 digits?
6000
7500
30,000
60,000
75,000
![UKMT (UK Mathematics Trust): [√ n]](https://images.squarespace-cdn.com/content/v1/580f3dddc534a5c144fe8a58/1600772690017-AF5AAN2C2RCFEV33MKQF/Screen+Shot+2020-09-22+at+11.24.25.png)
The notation [√ n] means the integer part of the square root of n. How quickly can you solve?
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Take away two matches to leave two squares.
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It is extraordinary that so much of the universe can be explained using mathematical equations. Indeed, it is often said that mathematics is the language of the universe. It is certainly the language of science.
In this clip, from a documentary in the American Nova series, the jazz musician Esperanza Spalding explains how maths is also at the heart of music. Pay attention to the way that numbers relate to musical intervals (an octave, a fifth and a fourth).
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Shawn has dealt four cards in front of you. He claims that if a card has an even number on one side of it, then the other side of the card is blue. Which cards do you need to turn over, in order to confirm if Shawn is telling you the truth?
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Shearing is a transformation of a shape in which a particular line (in this case the base of the triangle or parallelogram) remains fixed and all other points in the shape are translated parallel to that line by an amount proportional to the distance from that line.
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A cone shaped iceberg with tip pointing up is 80% submerged in water. Part I can see is 8m high & 6m across. What is height of full iceberg?
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2. Is there a link between the number of squares in a polynomial (a shape made out of squares joining on a complete edge) and the minimum perimeter that can be made for each number of squares?
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The Ancient Egyptian Rhind Papyrus dates back to around 1550 BC and is full of mathematical ideas and puzzles. Not surprisingly, it has a section about calculating the slopes of pyramids.
I particularly like that it opens with the statement: “Directions for Attaining the Knowledge of All Dark Things”. That’s quite a way to describe mathematics.
The papyrus has a great deal about “Egyptian fractions”, which means that every fraction has to be described in terms of other fractions which have the numerator 1.
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Can you solve?
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Alex has 25 discs that are black on one side and white on the other. He arranges them on a 5 x 5 grid so that all the white sides are showing.
A move consists of taking any three consecutive discs in a row or column and flipping them over. You want the discs to make the checkerboard coloring shown below.
From the initial all-white position, what is the smallest number of moves needed to get to the checkerboard position?
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UKMT volunteer Fraser Heywood kindly shared this video of his favourite mathematical puzzle with us. The Tower of Hanoi is a mathematical puzzle and a recursive algorithm, where the objective is to move an entire stack of disks from the source position to another position.
The three rules are:
- Only one disk can be moved at a time.
- A disk can only be moved if it is the uppermost disk on a stack.
- No larger disk may be placed on top of a smaller disk.
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