Sunday Afternoon Maths IV

Here's this week's collection. Answers & extensions can be found here. Why not discuss the problems on Twitter using #SundayAfternoonMaths. This Tuesday is Maths Jam, although I won't be there as it is also parents' evening at my school.

Arccos + Arcsin

Source: EdExcel A-level C3 paper January 2007

What is the value of arccos(x) + arcsin(x)?

i ⁱ

What is the value of i to the power of i?

x^xxxxx...

If x^xxxxx... [x to the power of (x to the power of (x to the power of (x to the power of (...)))) with an infinite number of xs] is equal to two, what is the value of x?

4 and 5 digit numbers

Source: UKMT Grey Kangaroo 2013

Adam chooses a 5-digit positive integer and deletes one of its digits to form a 4-digit integer. The sum of this 4-digit integer and the original 5-digit integer is 52713. What is the sum of the digits of the original 5-digit integer?

Sunday Afternoon Maths III Answers & Extensions

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

Colliding Parallel People

By walking in a straight line, the people will follow a great circle. They begin by travelling parallel to each other, so the 1km they are apart at the start is the furthest they will be apart. They will be 1km apart again after walking half way around the world and will collide at one quarter and three quarters of the way around.

Hence, each person will walk one quarter of the circumference of the Earth before colliding, which will be 10,018km.

Another way to see that this is the answer is to assume that the people stand on the equator and walk North. They will meet at the North Pole, one quarter of the way round.

Extension

If two people stand 10km apart and walk in the same direction, how far will the have to walk until they collide due to the curvature of the Earth? (diameter of Earth = 12,742km)

Twenty

There are two ways to make 20 by multiplying three digits: 2×2×5 and 1×4×5. Listing all the possible orderings of these, we have:
145
154
415
451
514
541
225
252
522
Therefore, there are 9 different three digit numbers where the product of the digits is 20.

Extension

How many 4 digit numbers are there where the product of the digits is 20?
5 digit?
n digit?

Sunday Afternoon Maths III

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

Colliding Parallel People

If two people stand 1km apart and walk in the same direction, how far will the have to walk until they collide due to the curvature of the Earth? (diameter of Earth = 12,742km)

Twenty

Source: UKMT Junior Mathematical Olympiad 2013

How many three digit integers are there for which the product of the digits is 20?

Sunday Afternoon Maths II Answers & Extensions

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

Guess the Diagonal

CG is equal to AF as they are both diagonals in a rectangle.
AF is a radius of the triangle, so AF = 10.
Therefore CG = 10.

Extension

Calculate the lengths of CI and CJ.

Calculate the length of HG.

Sunday Afternoon Maths II

Here's this week's collection. Solutions will be posted on Monday.

Guess the Daigonal

My Best Mathematical and Logic Puzzles by Martin Gardner

Calculate the length of the diagonal CG.

Sunday Afternoon Maths I Answers & Extensions

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

Burning Ropes

Light one rope at both ends and the other at one end. When the first rope finishes burning, light the other end of the second rope. The second rope will finish burning 45 minutes after the start.

Extension

What are all the possible times you can measure with two ropes? How about three ropes? Four ropes? n ropes?

Pole Position

The two poles can be any distance apart; the distance does not affect the heights.

Extension

If the heights of the two poles were 12m and 24m tall, what height would the intersection of the lines be?
If the heights of the two poles were am and bm tall, what height would the intersection of the lines be?

Circles

Call the blue area B and the red area R.
R = Area of quarter circle - Area of semicircle - (Area of semicircle - B)
R = ¼π(2r)² - ½πr² - (½πr² - B)
R = πr² - ½πr² - ½πr² + B
R = 0 + B
R = B

Extension

Prove that the red and blue areas are equal.

Pyramid and Tetrahedron

Let the length of a side of a triangle be 2L (I used 2L and not L to get rid of fractions in the calculations).
By Pythagoras' Theorem, the height of a triangle is L√3.
Using Pythagoras' Theorem again, the height of the square-based pyramid is L√2.
Therefore, the volume of the square-based pyramid is ⅓×(2L)²×L√2.
This simplifies to ⁴/₃√2L³.

Next, we find the area of the tetrahedron.
Call the point on the base of the tetrahedron, directly below the vertex at the top A.
Using cosine in the triangle made by A, the corner of the base and the midpoint of a side of the base, the distance from the corner to A is ²/_√3L.
Using Pythagoras' Theorem yet again, we find that the height of the tetrahedron is L^2√2/_√3
Therefore, the volume of the tetrahedron is ⅓×½×2L×L√3×L^2√2/_√3.
This simplifies to ²/₃√2L³.

Finally, the ratio of volume of the square based pyramid to the tetrahedron is:
⁴/₃√2L³ : ²/₃√2L³
2 : 1

Extension

What would the ratio be if they were isosceles triangles?

8! minutes

8×7×6×5×4×3×2×1 minutes
  = 8×7×4×3×1 hours (dividing by 60)
  = 7×4×1 days (dividing by 24)
  = 4×1 weeks (dividing by 7)
  = 4 weeks

Extension

8 is the smallest number n such that n! minutes is a whole number of weeks.
What is the smallest number m such that m! seconds is a whole number of weeks?

Sunday Afternoon Maths I

Following some relatively popular Twitter posting of maths problems, I've decided to start posting a weekly collection of interesting puzzles I have encountered. I'll be posting solutions on the following Monday.

Here's this week's collection, including puzzles from this month's MathsJam:

Burning Ropes

Source: Richard Wiseman's Blog

You have two ropes and some matches. Each rope, if lit at its end, will burn for 60 minutes. But the rate of burning is not regular, so cutting a rope in half doesn’t result in a burn time of 30 minutes. How can you use the ropes to time exactly 45 minutes?

Pole Position

Source: Futility Closet

Two poles stand vertically on level ground. One is 10 feet tall, the other 15 feet tall. If a line is drawn from the top of each pole to the bottom of the other, the two lines intersect at a point 6 feet above the ground. What’s the distance between the poles?

Circles

Source: MathsJam

There is a quarter circle with radius 2r and centre A and two semi circles with radius r and centres B and C.

Prove that the red area is equal to the blue area.

Pyramid and Tetrahedron

Source: MathsJam

If four equal equilateral triangles form the sides of a square-based pyramid, what is the ratio of the volume of the pyramid to the volume of the tetrahedron whose sides are the four triangles?

8! minutes

Source: UKMT Intermediate Challenge

How many weeks are there in 8! (8×7×6×5×4×3×2×1) minutes?