Solutions

Puzzle 1 (2 circles and square on tangent)

Interestingly there were 4 different solutions to this problem.

Mark 2ts

Constructed triangle AJE and solve the quadratic for its sides

David noticed that this triangle must be a 3, 4, 5 triangle so no need to solve a quadratic equation (in fact any right angled triangle of the form x, x-y, x-2y must be 3, 4, 5), so we set the challenge of solving this problem in other ways without solving a quadratic equation. This generated 3 other solutions.

Mike P

In the diagram O is the centre of the right hand circle, A is the mid point of the lower side of the square. B is the top left corner of the square and C is the point lying on the circle and on the perpendicular line from O.

Find the horizontal distance between the point B and the centre of the circle, O.

Let the angle AOC = θ = atan(1/2). BC had length 2 r cos θ. The horizontal distance between ) and B is 2 r cos θ sin θ = 2 r (1/√5) (2/√5) = 4 r/5.

The length of the square’s side is thus 2 r/5

Peter S

Brian A

Tangent secant theorem

Puzzle 2 (Set of Christmas puzzles)

Mathematician Shares Solutions For 10 Festive Brain Teasers

Puzzle 3 (Normal to circle diameter)

Puzzle 4 (3 switches)

Turn on the first two switches. Leave them on for five minutes. Once five minutes has passed, turn off the second switch, leaving one switch on. Now go through the door. The light that is still on is connected to the first switch. Whichever of the other two is warm to the touch is connected to the second switch. The bulb that is cold is connected to the switch that was never turned on.

Puzzle 5 (Pacific + Baltic + Arctic)

P A C I F I C B A L T I C A R C T I C C C C C C C C

+ + =

4658985 261285 635285 5555555

+ + =

Nathan then entered this problem into ChatGPT. “The response was reassuring (for the moment). It starts out fine but then loses it. It didn’t manage even when I suggested all the correct values”

I then mentioned this to my son and son-in-law.

• One put the same problem into the paid version of ChatGPT. This got the correct answer but took a long time, it seemed like several minutes.
• The other put the problem into Claude which wrote a program to solve it by brute force, trying all possible combinations. It also got the correct answer but was not fast.

Seems like we can do better than AI on this problem.

Puzzle 6 (Height of man)

Depends on how much alcohol he has consumed.
However I think he would be 2.4m tall. Big man.

Puzzle 7 (Perimeter of graph)

(13 + 12+ 11 + 5 + 6) – (7 + 16 + 3) + 4 = 25

Puzzle 8 (Sum of last digits)

7000

Write down the first few $d(n)$. They repeat after $d(20)$. Actually repeat after $d(10)$ but in reverse order. So $g(2000) = 100 g(20) =100 \times 70 = 7000$

$n$	$f(n)$	$d(n)$	$g(n)$
1	1	1	1
2	3	3	4
3	6	6	10
4	10	0	10
5	15	5	15
6	21	1	16
7	28	8	24
8	36	6	30
9	45	5	35
10	55	5	40
11	66	6	46
12	78	8	54
13	91	1	55
14	105	5	60
15	120	0	60
16	136	6	66
17	153	3	69
18	171	1	70
19	190	0	70
20	210	0	70

Puzzle 9 (Distance tangent to diameter)

Mike P Solution

Nathan Solution

Puzzle 10 (Set of integers, what’s not in set)