Puzzle of the Week #219 - Four Leaf Clover

Four identical pieces forming a square with a four-leaf-clover shaped hole can be rearranged (after flipping half the pieces) into a rhombus with three circular holes. The shorter diagonal of the rhombus is the same as the side length of the square.

 Rounded to the nearest whole number percentage, what proportion of the square is shaded?

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Puzzle of the Week #218 - Fast Flowing River

Grace and Ruby live in a house next to a fast flowing river. One day they decide to travel a few miles down the river and then return home. Grace is on foot, jogging along the riverbank at a speed of 5mph. Ruby takes the boat, which can travel at 9mph relative to the current of the river. They both set off at the same time, turn around at the same spot and arrive home at the same time. How fast was the current of the river?

 

Puzzle of the Week #217 - Identical Twins

Phyllis and Dilys are identical twins. They are each independently given the same 4-digit number.

Phyllis takes the number and converts it from decimal (base 10) to base 4, and writes down the 6-digit result.

Dilys simply writes the first and last digits of the number followed by the number in its entirety.

They are astonished to find that they have both written down the same 6-digit number!

What was the original number?

 

In other words, which number ABCD, when converted from decimal to base 4 becomes ADABCD?

 

 

Puzzle of the Week #216 - Four Semicircles in a Rectangle

Four semicircles are arranged around the edges of a rectangle. What are their respective radii?

This is a great deal more difficult than ‘Three Semicircles in a Square’. If it helps I can tell you that there is exactly one solution, and that each of the radii are rational numbers (so if you find it useful – and I’m not at all sure you will - you can scale the whole thing up by some factor, solve as a Diophantine set of equations, and scale back down again). I myself found this very difficult to solve, so if you discover a route to the solution that is not too difficult, I would be interested in knowing it.

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Puzzle of the Week #214 - Travelling Salesman

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A travelling salesman needs to visit each of the black squares in turn and return to where he started. He can choose to start wherever he wishes. He traces a zero-thickness path, can travel at any angle (this gridlines are only a reference) and doesn’t need to go to the middle of each black square, it is sufficient for his route to touch a corner or an edge. If each small square measures 1 x 1, what is the length of the shortest route?

Puzzle of the Week #213 - Collate

Given a string with three each of three different letters ordered thus:

 

A A A B B B C C C

 

It is possible through a series of three moves to change the order into:

 

A B C A B C A B C

 

A ‘move’ consists of taking a section of the string and reversing the order of the items within it. The brackets show the section to be reversed in the subsequent move:

 

A(AABB)BCCC -> ABBA(ABC)CC -> AB(BACBAC)C -> ABCABCABC

 

Using the exact same idea, how many moves will it take to change between the following?

 

A A A A A B B B B B C C C C C D D D D D E E E E E

 

A B C D E A B C D E A B C D E A B C D E A B C D E

 

Puzzle of the Week # 212 - Infinite Sum

The convergent sum of the following infinite sequence, in its simplest terms, is a fraction with a square number on the top and a factorial number below.

 

1/7 + 1/16 + 1/27 + 1/40 + 1/55 + 1/72 + … + 1/n(n+6) + …

 

What is it?

 

Footnote: the partial sum converges extremely slowly, such that if you add the first million terms you will only get the first 5 decimal places. However there exists a very nice trick to allow you to work out exactly the number the infinite sum converges upon (without having to do an infinite number of calculations!).

Puzzle of the Week #210 - Return Journey

The towns of Abbottsville, Beresford and Christchurch all lie on one straight road. I embark on a journey from Abbottsville, through Beresford, to Christchurch, and back the same way. Each day I cover 1 mile more than the day before. It takes me 10 days to travel from Abbottsville to Beresford, 11 days to travel from Beresford to Christchurch and 12 days to travel from Christchurch (through Beresford) to Abbottsville.

What are the distances between the towns?

Puzzle of the Week #209 - Coin vs Dice

Which of the following scenarios should on average* take the most throws:

Repeatedly tossing a coin until you have seen at least 6 heads and at least 6 tails, or

Repeatedly throwing a dice until you have seen each of the numbers 1 to 6 at least once each?

 

*(simple arithmetic mean)

 

Puzzle of the Week #206 - Tumbler

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A glass tumbler has an outside diameter of 78mm, and inside diameter of 72mm and a solid base that is 30mm deep.

Its centre of gravity is exactly 30mm from the table, ie, at the height of the bottom of the inside of the glass.

What is the overall height of the glass tumbler?

Puzzle of the Week #205 - Formula

I have a sequence. It begins:

1

101

10101

1010101

101010101

etc

As it’s populated entirely with 1s and 0s it is not entirely clear whether these are decimal numbers, or binary numbers, or some other base entirely.

 

Construct a formula to give the nth term, which works no matter what base we are working in.

***EDIT*** You are looking for a simple formula where you plug in the value of n and get the nth term, no recursive formulae or finite sums required.

Puzzle of the Week #203 - Fractions

1/3 = 0.333… and 3/8 = 0.375. No fraction with a single-digit denominator lies between them.

What is the fraction greater than 1/3 and less than 3/8 that has the lowest denominator? Call that fraction ‘x’.

What is the fraction greater than ‘x’ and less than 3/8 that has the lowest denominator? Call that fraction ‘y’.

Finally, what is the fraction greater than ‘x’ and less than ‘y’ that has the lowest denominator?

Puzzle of the Week #202 - Spherical Asteroid

A perfectly spherical asteroid is in orbit somewhere in the solar system. There is a small deposit of Unobtainium buried within it.

Scientists are able to scan the asteroid and determine the position of the deposit of Unobtainium relative to the surface of the sphere, by imposing a perpendicular coordinate system on it:

From the Unobtainium, if you go the 24 metres in the positive x direction you reach the surface

From the Unobtainium, if you go the 60 metres in the negative x direction you reach the surface

From the Unobtainium, if you go the 30 metres in the positive y direction you reach the surface

From the Unobtainium, if you go the 36 metres in the positive z direction you reach the surface

What is the radius of the spherical asteroid?

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This may help you: The general equation of a sphere is: (x - a)² + (y - b)² + (z - c)² = r², where (a, b, c) represents the centre of the sphere, r represents the radius, and x, y, and z are the coordinates of the points on the surface of the sphere.

Puzzle of the Week #201 - Two of One and One of Another

I recently investigated which numbers can be represented at the sum of one square number and double a different square number, for instance 9 is 1+2(4), or 38 is 36+2(1).

 

With pencil and paper: 99 is the smallest number that can be represented as a^2 + 2b^2 in three different ways: find them all!

 

With a computer: what is the smallest number that can be represented as a^2 + 2b^2 (with a and b different positive integers) in six different ways?