Sunday Afternoon Maths XII

Here's this week's collection. Answers & extensions tomorrow. Why not discuss the problems on Twitter using #SundayAfternoonMaths or on Reddit.

Circles

Which is largest, the red or the blue area?

Largest Triangle

What is the largest area triangle which has one side of length 4cm and one of length 5cm?

Unit Octagon

Source: UKMT Junior Mathematics Challenge 2014

The diagram shows a regular octagon with sides of length 1. The octagon is divided into regions by four diagonals. What is the difference between the area of the hatched region and the area of the region shaded grey?

Sunday Afternoon Maths XI Answers & Extensions

This post contains the answers to this week's Sunday Afternoon Maths and some extension problems based around the originals.

Light Work

Source: By Lewis Carroll via Futility Closet

Extension

If instead a window ³/₄ of the size was required, how could this be done?

Semi Circle in a Triangle

Label the triangle as follows:

EC and BC are both tangents to the circle so FEA is a right angle and the lengths EC and BC are equal, so the length of EC is 5. Let r be the radius of the circle.

Using Pythagoras' Theorem in triangle FEA:

$$8^2+r^2=(12-r)^2$$ $$64+r^2=144-24r+r^2$$ $$24r=80$$ $$r=\frac{80}{24}=\frac{10}{3}$$

Extension

What is the radius of the circle whose diameter lies on the 5cm side and to which the 12cm and 13cm sides are tangents?

What is the radius of the circle whose diameter lies on the 13cm side and to which the 12cm and 5cm sides are tangents?

Additional observation

For each pair of semi circles, draw a straight line between the two points where the semi circles intersect. These lines all meet at a point.

Sunday Afternoon Maths XI

Here's this week's collection. Answers & extensions here. Why not discuss the problems on Twitter using #SundayAfternoonMaths or on Reddit.

Light Work

Source: By Lewis Carroll via Futility Closet

I don’t know if you are fond of puzzles, or not. If you are, try this. … A gentleman (a nobleman let us say, to make it more interesting) had a sitting-room with only one window in it–a square window, 3 feet high and 3 feet wide. Now he had weak eyes, and the window gave too much light, so (don’t you like ‘so’ in a story?) he sent for the builder, and told him to alter it, so as only to give half the light. Only, he was to keep it square–he was to keep it 3 feet high–and he was to keep it 3 feet wide. How did he do it? Remember, he wasn’t allowed to use curtains, or shutters, or coloured glass, or anything of that sort.

Semi Circle in a Triangle

Source: UKMT Pink Kangaroo 2012

This right-angled triangle above has sides of lengths 12cm, 5cm and 13cm. The diameter of the semicircle lies on the 12cm side and the 13cm side is a tangent to the circle. What is the radius of the semi circle?

Sunday Afternoon Maths X Answers & Extensions

This post contains the answers to this week's Sunday Afternoon Maths and some extension problems based around the originals.

Three Digit Numbers

The sum of the products is:

$$\sum_{i=1}^{9}\sum_{j=1}^{9}\sum_{k=1}^{9}ijk$$ $$=(\sum_{i=1}^{9}i)(\sum_{j=1}^{9}j)(\sum_{k=1}^{9}k)$$ $$=45\times45\times45$$ $$=45^3$$

Extension

Find the sum of the products of the digits of all n digit numbers.

Princess Problem

Imagine a chessboard with 17 columns and a large number of rows. Let each column correspond to one of the rooms. In the first row, mark the room in which the princess is. Every day following, mark her location in the next row.

The princess will always move one square diagonally, so will always be on the same colour square she started on. Begin by checking the 17th room, then the 16th room, then continue down the the first room. If the princess started on a black square you will have found her, as she has no way of getting past you.

If you have not caught the princess, then she must have started on a white square. Checking rooms 17 down to one again will find her as this time it will start on a white square.

Extension

Is there a quicker way to find the princess?

Lights

If two lights opposite each other are on and the other two are off, then toggling two opposite lights will always lead to your release. It can be checked that the following combination will always lead to this:

• Toggle two opposite lights
• Toggle two adjacent lights
• Toggle two opposite lights
• Toggle one light
• Toggle two opposite lights
• Toggle two adjacent lights
• Toggle two opposite lights

Extension

Can the same be done with eight lights?

Sunday Afternoon Maths X

Here's this week's collection, including problems from last week's Maths Jam. Answers & extensions here. Why not discuss the problems on Twitter using #SundayAfternoonMaths or on Reddit.

Three Digit Numbers

Source: UKMT Pink Kangaroo 2012

Brigette wrote down a list of all 3-digit numbers. For each of the numbers on her list she found the product of the digits. She then added up all of these products. Which of the following is equal to her total?

A 45     B 45²     C 45³     D 2⁴⁵     E 3⁴⁵

Princess Problem

Source: Maths Jam

A princess lives in a row of 17 rooms. Each day she moves to a room adjacent to the one she wakes up in (eg. If she sleeps in room 5 today, then she will sleep in room 4 or 6 tomorrow). If you are able to find the princess by only opening one door each night then you will become her prince. Can you find her in a finite number of moves?

Lights

Source: Maths Jam

You have been taken captive and are blindfolded. There is a table in front of you with four lights on it. Some are on, some are off: you don't know how many and which ones. You need to get either all the lights on or all the lights off to be released. To do this, you can ask your captor to toggle the light switches of some of the lights. You captor will then rotate the table (so you don't know where the lights you toggled now are). Find a sequence of moves which will always lead to your release.

Sunday Afternoon Maths IX Answers & Extensions

This post contains the answers to this week's Sunday Afternoon Maths and some extension problems based around the originals.

Equal Areas

Let A be the area of the square (and the triangle).

The length of a side of the square is √A, so the perimeter of the square is 4√A

Let l be the length of a side the triangle. Then ¹/₂l ²sin60 = A, so l ² = ^4A/_√3. Therefore l = ^2√A/_3^1/4 and the perimeter of the triangle is ^6√A/_3^1/4.

Hence the ratio of the perimeters is ^6√A/_3^1/4:4√A which simplifies to 3^3/4:2

Extension

If an n sided regular polygon has the area A, what is the length of one of its sides?

Chessboard Squares

There are 64 1×1 squares, 49 2×2 squares, 36 3×3 squares, 25 4×4 squares, 16 5×5 squares, 9 6×6 squares, 4 7×7 squares and 1 8×8 square on a chessboard.

This can be shown by counting how many positions the top left corner of the square can sit on. For example, the top left corner of a 5×5 square can be in the first four rows and columns of the board (otherwise the square will go off the board) and 4×4=16.

64+49+36+25+16+9+4+1=204.

Extension

Source: Thinking Mathematically by John Mason, Leone Burton & Kaye Stacey

How many rectangles are there on a chessboard?

Downing Street

The probabilities can be summarised as follows:

	First person truthful	First person lying
Second person truthful	3/4×4/5=12/20	1/4×4/5=4/20
Second person lying	3/4×1/5=3/20	1/4×1/5=1/20

As they both agree, only both lying and both truthful are possible. Hence the chance of them lying is ¹/₁₃ and the chance of them telling the truth, and it indeed being the Minister of Maths, is ¹²/₁₃

Extension

If the first person said it was the Minister of English and the second said it was the Minister of Maths, what is the probability that it was the Minister of Maths?

Multiple Sums

The multiples of 3 less than 1000 are 3,6,9,...,999; the multiples of 5 are 5,10,15,...,995. Multiples of 15 (15,30,...,990) will appear in both lists so we are trying to find (3+6+9+...+999)+(5+10+15+...+995)-(15+30+...+990). This is:

$$\sum_{i=1}^{333}3i+\sum_{j=1}^{199}5j-\sum_{k=1}^{66}15k$$ $$=3\sum_{i=1}^{333}i+5\sum_{j=1}^{199}j-15\sum_{k=1}^{66}k$$ $$=3\times\frac{333\times334}{2}+5\times\frac{199\times200}{2}-15\times\frac{66\times67}{2}$$ $$=166833+99500-33165$$ $$=233168$$

Massive thanks to MathJax for allowing me to typeset maths!

Extension

Find the sum of all the multiples of 3 or 5 below n.

Sunday Afternoon Maths IX

Here's this week's collection. Answers & extensions here. Why not discuss the problems on Twitter using #SundayAfternoonMaths or on Reddit. This Tuesday is Maths Jam, make sure you're there!

Equal Areas

An equilateral triangle and a square have the same area. What is the ratio of the perimeter of the triangle to the perimeter of the square?

Chessboard Squares

Source: Thinking Mathematically by John Mason, Leone Burton & Kaye Stacey

It was once claimed that there are 204 squares on a chessboard. Can you justify this claim?

Downing Street

Source: My Best Puzzles in Mathematics by Hubert Phillips

A knot of spectators in Downing Street was watching members of the Cabinet as they arrived for a critical meeting.

"Who's that?" I asked my neighbour, as a silk-hatted figure, carrying rolled umbrella, rang the bell at No. 10. "Is it the Minister of Maths?"

"Yes," he said.

"Quite right," said a second spectator. "The Minister of Maths it is. Looks grim, doesn't he?"

The first of the speakers tells the truth three times out of four. The second tells the truth four times out of five.

What is the probability that the gentleman in question was in fact the Minister of Maths?

Multiple Sum

Source: Project Euler

If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.

Find the sum of all the multiples of 3 or 5 below 1000.