Solution of the Week #518 - Four Rectangles

In the general case of two rectangles overlaid such that they have one corner in common and each has a corner coinciding with the side of the other rectangle, it will produce a pair of similar triangles as below. From this we can tell that C/A=B/D. Cross-multiplying shows that AB=CD, in other words the two rectangles have the same area. So in the original figure, all four rectangles have the same area. We can’t know the exact length and width of the red and blue rectangles, but since the green and magenta rectangles also share an edge, that gives us enough information to solve the puzzle. If we call the height of the magenta rectangle ‘x’, then the height of the green rectangle is x-14.

36*(x-14)=27x

9x=36*14

x=56

Solution of the Week #516 - Two Pairs of Circles

The answer is that the radius of the incircle is 7. I only really know that because I already knew that when I constructed the puzzle. I have an extremely messy and complicated way to find it only given the information in the question, which I won’t bore you with, but if you found it in a neat way please do let me know.

Edit: see comments on the puzzle page, neat solutions to be found there.

Solution of the Week #515 - Three Circles in a Trapezoid

Following the previous puzzles and the relationships between lengths of a right trapezium with an inscribed circle, if we were to follow that through to equations I believe we would end up with quartic equations awkward to work with.

More practically in this instance we can just use trial and error, and very quickly find that each trapezoid as we descend the figure is 50% bigger than the one above. So the horizontal lines measure 40, 60, 90 and 135. The sloping lines will then be 51, 78 and 117. And the vertical lines will be 48, 72 and 108, giving the overall height of 228.

Solution of the Week #514 - Two More Circles in Trapezoids

If we call the length of the middle horizontal x, we can use the formula for the radius of the incircles as before to determine the height of each trapezoid in terms of x.

Since we are told the angle on the right is a right angle, we can find two similar triangles above and below it (in fact they are congruent, but we don’t need to assume that for our solution).

So (x-5)(x-12)=10x/(x+5) * 24x/(x=12)

(x+5)(x-5)(x+12)(x-12)=240x^2

(x^2-25)(x^2-144)=240x^2

Let’s make a substitution and call x^2 y

(y-25)(y-144)=240y

y^2-409y+3600=0

y=9 or 400

x therefore is +/-3 or +/-20

In the diagram, it is clearly bigger than 12, so must be equal to 20.

Plugging this value into the height expressions above gives the overall height as 8+15=23.

Solution of the Week #513 - Two Circles in a Trapezoid

The trivial answer is if A=B=C=D, wherein A/B = 1.

To find the other solution, first consider that the top part of the shape is similar, in the mathematical sense, to the bottom part. Let’s say that the top horizontal has length 1, then the middle horizontal has some unknown factor x, and the bottom horizontal will therefore be x^2.

Next let’s look at the properties of a right trapezoid with a circle inscribed within it.


Splitting the lengths at the tangent points we call the radius of the circle R and the other two lengths S and T, forming a right angled triangle to combine them.

Using Pythagoras, (T+S)^2-(T-S)^2=(2R)^2, therefore ST=R^2.

Now let’s define S as U-R and T as V-R.

(U-R)(V-R)=R^2

UV-R(U+V)+R^2=R^2

UV=R(U+V)

R=UV/(U+V)

Therefore we can find the radius of a circle within a right trapezoid as simply the product of the top and bottom sides divided by their sum. The height of the trapezoid will be twice that.

And for completeness, the sloping side will be the sum of the top and bottom sides, less the height of the trapezium.

 

Let’s now look at the top trapezoid in the puzzle, where we have defined the top edge to be 1 and the bottom edge to be x.

The left side will be 2x/(1+x). That denominator is awkward, so let’s scale everything up by (1+x). We can do that because ultimately it is a ratio between sides that we are seeking.

Now remember everything in the bottom trapezoid is the same as the top one, but scaled up by x. We can therefore find expressions for A, B, C and D in terms of x:

 

A=2x^2

B=x+1

C=2x

D=x^3+x

 

We are told that A+C=B+D

2x^2+2x=x^3+2x+1

x^3-2x^2+1=0

 

We know that one solution (the trivial solution) is that x=1, which leaves the quadratic x^2-x-1=0. This has one positive solution: (1+sqrt(5))/2, or the golden ratio.

Finally, plugging this into the expressions for A and B gives a ratio for A/B of exactly 2.

 

So the solution is A/B = 2.

If we rescale back to letting B=1, A=2, C will be sqrt(5)-1 and D is sqrt(5). The diagonal line will have a gradient of -2. And of course the middle horizontal will be equal to the golden ratio.

 

 

 

Solution of the Week #513 - Two Circles in a trapezoid

The trivial answer is if A=B=C=D, wherein A/B = 1.

I couldn’t find an elegant way to get to the other solution, so I won’t attempt to present one. I’ll just tell you that the solution is A/B = 2.

If we let B=1, A=2, C will be sqrt(5)-1 and D is sqrt(5). The diagonal line will have a gradient of -2. The middle horizontal will be equal to the golden ratio.

Solution of the Week #512 - 4x4x4 Cube

The argument for the 3x3x3 version hinges on the fact that the central cube will have 6 newly cut faces, and therefore will require 6 slices to form, regardless of rearrangement. For the 4x4x4 cube, there are 8 inner cubes, each of which need 6 slices directly acting on them. This can be done as long as the first cut in each direction is the middle one, and then the two halves can be lined up to simultaneously perform the other cuts in that direction. So perhaps counter-intuitively, a 4x4x4 cube also only needs 6 slices.

Solution of the Week #511 - Seven Circles

They will have a radius of 2. The base of the triangle will be 40, but the other sides cannot be determined.

17 circles of radius 1 will fit along the baseline.

 In general I have found that if one circle has radius A and two circles have radius B, then n circles will have radius:

 r_n = AB/((n-1)A-(n-2)B)

 In our case A=5, B=4. When n=7, r_7 = 20/(30-20) = 2.

 Reverse engineering r_n = 1:

 1 = 20/(5(n-1)-4(n-2)) = 20/(n-5+8)

20 = n-5+8

n = 17

Solution of the Week #510 - Pair of Circles 2

As before, the largest possible pair of circles will be tangent to the longest side of the triangle: the 15 side.

Using Heron’s formula we can find that the area of the triangle is 84.

From this we can work out that the inradius of the entire triangle will be 4.

The version with two circles will effectively be a scaled down version of the full triangle with one incircle, split apart and a 2r wide strip placed between.

The ratio of the inradius to the base length is the same for this split-apart triangle as it is for the full triangle.

 

(15-2r)/r = 15/4

60-8r = 15r

60 = 23r

r = 60/23