Solution of the Week #505 - Linked Values

The angles between the sides will not change as a and b change, since it is always an isosceles triangle with sides in the ratio 2,2,1. I have dropped an altitude to the midpoint of the base, giving us a right-angled triangle, whose smallest angle is x.

sin(x) = (ab/4)/ab = 1/4.

By Pythagoras the length of the altitude is ab*sqrt(15)/4.

cos(x) = ab*(sqrt(15)/4)/ab = sqrt(15)/4.

To find cos(2x) we can use the double angle identity:

cos(2x)=(cos(x))^2-(sin(x))^2 = 14/16 = 7/8.

Now focusing on the upper right triangle that has angle 2x, hypotenuse a(b-1) and adjacent side b(a-1).

cos = adj/hyp

7/8 = b(a-1)/(a(b-1))

7a(b-1) = 8b(a-1)

7ab-7a = 8ab-8b

8b-ab = 7a

b(8-a) = 7a

b = 7a/(8-a)

7a+ab = 8b

a(7+b) = 8b

a = 8b/(7+b)

Let’s see what happens when we let b increase. As b becomes massive the 7 in the denominator will become insignificant, and a will get closer and closer to 8, but never quite reach it, so an open upper bound for a is 8.

ab must be greater than a for the line across the triangle to be within the triangle, therefore an open lower bound for b is 1. This corresponds to an open lower bound for a of 1 also.

So the integer values that a could take are 2,3,4,5,6 or 7. Calculating the corresponding values of b, it is only an integer when a is 4, 6 or 7

The three integer solutions are a=4,b=7; a=6,b=21; a=7,b=49.

Solution of the Week #504 - Sum and Product

By Pythagoras the long side of the triangle is 87. Since a+b is 100 we can change b to 100-a. The small triangle at the top shares an angle with the large triangle, and since both are also right-angled the two triangles are similar.

87/63 = (63-a)/(a-13)

87(a-13) = 63(63-a)

87a-1131 = 3969-63a

150a = 5100

a = 34

b = 100-a = 66

a*b = 34*66 = 2244

Solution of the Week #503 - Semicircle in a Triangle

If we split the triangle into two triangles by drawing a line from the apex to the centre of the semicircle, we can see that the ‘height’ of each triangle is equal to the radius. So the total area of the triangle is 87r/2 + 75r/2 = 81r

 We can also calculate the area of the triangle using Heron’s formula, which states that the area is equal to the square root of the product s(s-a)(s-b)(s-c), where a, b and c are the sides of the triangle and s is the semi-perimeter.

 s=(87+75+108)/2 = 135

s(s-a)(s-b)(s-c) = 135*48*60*27 = 10497600

Area = sqrt(10497600) = 3240

 Now we have two different expressions for the triangle’s area we can equate them:

81r = 3240

And then find the radius

r = 3240/81 = 40. 

Solution of the Week #502 - Almost a Square

Here are the solutions. I have placed within each triangle what the primitive form of that Pythagorean triangle is.

Solution of the Week #501 - Rectangle and Quarter Circle

The first thing we can do is to work out the radius of the quarter circle. We can form a right triangle on the figure that has hypotenuse r and legs 6 and (r-2).

Using Pythagoras this works out as r=10.

If we make a copy of the figure, rotate it 90 degrees and place it next to the original figure we can see what the length and width of the rectangle are in terms of the radius.

Since we already know r=10, we find that the rectangle measures 14 x 4 units.

Solution of the Week #500 - Rectangle

If we call the length of the rectangle ‘x’ and the height of the rectangle ‘y’, then the value we are seeking is 2x+2y-xy.

If we just look at one half of the figure, replacing the circle with some radii and showing tangent lines as equal we get:

We now have a right-angled triangle with sides x, y and (x+y-2).

x^2+y^2 = (x+y-2)^2

x^2+y^2 = x^2+2xy-4x+y^2-4y+4

Subtract x^2+y^2 from both sides:

0 = 2xy-4x-4y+4

Subtract 2xy-4x-4y from both sides:

4x+4y-2xy = 4

Divide all terms by 2

2x+2y-xy = 2

The value of 2x+2y-xy is what we are seeking, and so the answer is 2. There isn’t enough information to determine x and y, but the difference between the perimeter and the area will be constant.

Solution of the Week #499 - Octagon to Square

The above shows how the dissection would be achieved. If we call the side length of the octagon s and the radius of the small circles r, then we can say that an octagon with side s, subtract an octagon with side 2r must have the same area as a square with side 2s.

The formula for the area of a regular octagon is (2+2sqrt2)s^2. The area of the inner octagon is 4(2+2sqrt2)r^2, and the area of the full square is 4s^2.

 (2+2sqrt2)s^2 - 4(2+2sqrt2)r^2 = 4s^2

 Let’s use this to find a relationship between s and r.

 r = s/2(sqrt(3-2sqrt2))

 If we let the square be a unit square, s is 1/2 and therefore r is (sqrt(3-2sqrt2))/4. The area of each of the circular holes is pi*r^2 = pi*(3-2sqrt2)/16.

 The shaded part of the square is 1 less the area of five holes,

1-5*pi*(3-2sqrt2)/16

which is approximately equal to 0.83156…

So the square is about 83.156% shaded.

Solution of the Week #498 - Three Week Odyssey

Say I walked x miles on the first day. In the first week I will have walked 7x+21 miles. In the second week, 7x+70 miles. In the third week, 7x+119 miles.

These distances need to form a right angles triangle, so we can use Pythagoras Theorem:

(7x+21)^2 + (7x+70)^2 = (7x+119)^2

49x^2+294x+441 + 49x^2+980x+4900 = 49x^2+1666x+14161

49x^2-392x-8820=0

x^2-8x-180=0

(x+10)(x-18)=0

So the distance travelled on the first day was either -10 miles or 18 miles. Since the distance needs to be positive the answer is 18 miles. In total I walked 147+196+245 = 588 miles.

A shortcut to the solution would be to note that the difference between week 2 and week 1 is the same as the difference between week 3 and week 2: 49. So we are looking for a scaled up version of a primitive Pythagorean triple in arithmetic progression. Since 3,4,5 is the only such triangle we need only scale this up by a factor of 49 to find our distances.

Solution of the Week #497 - Six Regions

Here are the coordinates of all of the junctions. The marked point has coordinates of (588,1064).

Solution of the Week #496 - Four Towns

Since Gunton is due north of Kipton and Lawton is due east, Gunton-Kipton-Lawton form a right angle. Since we don’t know the speed we can’t yet directly equate the given times and the given distances. Let’s say we are travelling at ‘s’ miles per minute. We can create the following diagram:

We can use the law of cosines to give an expression for cos(a) and cos(90-a) in terms of s:

 Cos(a) = (15^2+(78s)^2–(102s-15)^2)/(2*15*78s)

Cos(a) = (17-24s)/13

Cos(90-a) = (16^2+(78s)^2–(82s-16)^2)/(2*16*78s)

Cos(90-a) = (41-10s)/39

 But, cos(90-a) is also expressible as sin(a), and (sin(a))^2+(cos(a))^2=1

 (51-72s)^2+(41-10s)^2 = 39^2

5284s^2-8164s+2761=0

 s can be 1/2 or 2761/2642

 If we use this second figure and plot the resulting distances, Torton would be way to the north-west, not generally north-east as stated, so the speed is 1/2 mile/minute or 30mph. The distance between Kipton and Torton is therefore 39 miles.

Solution of the Week #495 - Smallest Triangle

If we call the base b, and the other sides a and c, such that a>c (a can’t be equal to c as the mirror line would be parallel with the base). We want a and c to be as close as possible, so that the angle of the mirror line is as shallow as possible. Since they are integers the smallest difference is 1, so let’s say a-1=c. With that being the case the length that we want to be 45 will be bc (in general it would be bc/(a-c)). To minimise the perimeter we want b to be roughly double c. If we had said we wanted the dashed line to be 50 instead of 45, we could achieve that exactly with b=10, c=5 and a=6.

Here the closest is b=9, c=5, a=6, with a perimeter of 20, and area of 10*sqrt(2).

Solution of the Week #494 - Five Dice

The fact that none of the rearrangements are even means that each individual roll is odd.

The fact that we are told that b>d and we are expected to figure out the entire sequence must mean that there are four rolls of one number and one roll of a second number.

The gives few enough possibilities for trial and error. We need to try each of five rearrangements of 13333, 15555, 31111, 35555, 51111, 53333 for divisibility by 7.

The only one that is never divisible by 7 is 35555, therefore the sequence of dice rolls was 55535.

For completeness, the sequences that are never divisible by 7 are:

11111 and its multiples 22222 33333 44444 55555 66666

11112 and its multiples 22224 33336

The result of subtracting those three from 77777: 66665 55553 44441

Every other sequence of dice rolls can be arranged to be a multiple of 7.

Solution of the Week #493 - Grid Fill 3

As we are told that there are 5 black squares that limits the possible arrangements. Since each of the four edges of the grid must contain an even number of black squares, they must appear in an odd number of corners. That leaves just seven distinct arrangements of black squares: four with one black corner and three with three black corners. One of the seven arrangements of black squares leads to a symmetrical solution. Five more arrangements don’t lead to any solutions.

Here is the asymmetrical solution:

And if you’re interested, here are the other 18 solutions I found:

Solution of the Week #492 - Grid Fill 2

These are (I believe) the only five solutions for a 6x6 grid.

Solution of the Week #491 - Grid Fill

These are (I believe) the only distinct solutions for a 7x7 grid.

In general, a solution can always be found that involves one large rectangle or square which contacts all four sides of the grid. Neither of the edges of this rectangle can be a multiple of 3, hence why for the first solution, the main rectangle is 2x5, as 2 and 5 are the only pair of numbers adding to 7 that don’t involve a multiple of 3.

Solution of the Week #490 - Inheritance

Let’s start with the brothers having equal amounts, work backwards to the will, and forwards again to the alternative situation.

Let’s say they ended with £x each. The final round is Adam’s, so immediately prior to that they would have:

A 2x

B x/2

C x/2

Then Barnaby’s share is doubled and the other two decrease by the 1/2 of what Barnaby had:

A 7x/4

B x

C x/4

Then Charles:

A 13x/8

B 7x/8

C x/2

To check we haven’t gone wrong, at each stage the amount should add up to 3x, since we are only moving money between the brothers.

Next we need to move forward from this point with the alternative order.

First Adam splits his share such that his share is halved and 1/4 of what Adam had is added to each of the others:

A 13x/16

B 41x/32

C 29x/32

Then Barnaby splits his share:

A 145x/128

B 41x/64

C 157x/128

And finally Charles:

A 737x/512

B 485x/512

C 157x/256

We are told that in this scenario Adam ends up with £19881 more than Charles, so:

737x/512 – 157x/256 = 19881

x(737-314) = 19881*512

x = 24064

Then going back to the will scenario (13x/8 etc) we can say that the will granted the following amounts:

Adam: £39,104

Barnaby: £21,056

Charles: £12,032

Solution of the Week #489 - Triple Coin Flipping

It turns out that the following arrangement, using only ten coins, can always be solved. Not coincidentally, there are ten possible triplets of coins to flip (four vertical, two horizontal, and four diagonal).

Solution of the Week #488 - Tournament

Firstly the fact that the matches are drawn between rounds as opposed to the entire draw being set at the start of the tournament is immaterial. To reach the final, team 5 needs to be the highest ranked team on their side of the draw.

So the chance that the other seven teams in the same side of the draw are ranked lower than fifth is:

11/15*10/14*9/13*8/12*7/11*6/10*5/9

This cancels down neatly to 2/39, which is just over 5%.

Alternatively, you could ensure that the best four teams are all in the other side of the draw to team 5:

8/15*7/14*6/13*5/12

Which again cancels down to 2/39.

Solution of the Week #487 - How Many Ways

Obviously the final digit needs to be a 5. For the first two digits to be divisible by 3, they simply need to sum to a multiple of 3. Without the 5 there are three ways (1,2; 2,4; 3,6) and their reversals.

For the third and fourth digits to make a number divisible by 4, either the fourth digit is a multiple of 4 and the third digit is even, or the fourth digit is an even number not divisible by 4 and the third digit is odd. The fifth digit is whatever is left. Since at least one of the remaining three numbers will be odd, we have the same situation as before where if one arrangement is possible, a second is also possible. (See POTW #484).

So overall we have 3 pairs for the first two digits, doubled as they can be reversed, and doubled again for the remaining three digits.

12 possibilities in all:

126435

123645

216435

213645

241635

243615

421635

423615

361245

362415

631245

632415

Solution of the Week #486 - Not a Square

I start just over 12.5 miles from the south pole, let’s say 12.54 miles.

After walking south I am 2.54 miles away from the pole.

Walking east I am still 2.54 miles from the pole, but have travelled around 225.7 degrees around it.

Heading north, directly away from the pole, I end up 12.54 miles from the pole.

Finally heading west I reverse the direction around the pole, but this time by about 45.7 degrees, ending up roughly opposite where I started, but around 25.08 miles from it.

In fact if I’d started a little more than 12.54 miles from the south pole my final position would be even further away, but would depend on the curvature of the earth, but even assuming local flatness it’s no more than about 25.23 miles.

The exact maths of ending up opposite where you started is that you would start 10+10*(sqrt(pi*(4+pi))-pi)/(2*pi)=12.53863… miles away from the South Pole. The distance between start and end are then just double that.

The overall maximisation is beyond my abilities.