Mathematical Quickies No.226

So for some reason I never started looking for substitutions when solving equations. It was certainly something I learned to do when analyzing circuits, but when I see a math problem I never started looking for substitutions that might simplify the problem. Until recently.

Looking at, I think this problem, it “just clicked” – and I started looking for substitutions. Then used them to nock out answers for the next half dozen puzzle problems I tackled. Weird, since I am not doing anything I did not know before, but I just started looking at problems differently.

This one is fairly straight forward:

Solve: (x-a)/b + (x-b)/a = b/(x-a) + a/(x-b)

It becomes way easier to solve after a simple substitution.
My Solution.

Simple tiling problem

This makes for a nice problem in tiling. At first glance it looks hard to solve – but you the solution is obvious seconds after you start formally trying to solve the problem. Put this as an example early in the chapter on tiling and segmentation problems to help motivate readers.

Simple tiling problem

Simple tiling problem

Find a buiried cable digging a trench

Problem nummber 110 from

There is a straight cable buried under a unit square field. You must dig one or more ditches to locate the buried cable. Where should you dig to guarantee finding the cable and to minimize digging? For example you could dig an X shape for total ditch length of 2*sqr(2) but there is a better answer.


Finding a solution  - page one

Finding a solution - page one


Finding a solution  - page two

Finding a solution - page two


Finding a solution  - page three

Finding a solution - page three

12 ball weighing problem – find the fake

Problem 85 from

In front of you are 12 pearls, 11 being real and one fake. The real ones all weigh the same and the fake one differs in weight from the real ones (may weigh more or less). With a balance scale and three weighings how can you weed out the fake one and determine whether it is too heavy or too light?

Twelve pearls, one fake, and a scale

Twelve pearls, one fake, and a scale

River crossing problem with predators

Problem 79 from

One one side of a river are three humans, one big monkey, two small monkeys, and one boat. Each of the humans and the big monkey are strong enough to row the boat. The boat can fit one or two bodies (regardless of size). If at any time at either side of the river the monkeys outnumber the humans the monkeys will eat the humans. How do you get everyone on the other side of the river alive?

River crossing with predators

River crossing with predators

Descending 200 meters with a 150 meter rope

Saw this problem at

A man is trapped atop a building 200m high. He has with him a rope 150m long. There is a hook at the top where he stands. Looking down, he notices that midway between him and the ground, at a height of 100m, there is a ledge with another hook. In his pocket lies a Swiss knife. Hmm… how might he be able to come down using the rope, the two hooks and the Swiss knife

Mixing red and blue paint – show the final ratio after a single mixing.

If you have two buckets, one with red paint and the other with blue paint, and you take one cup from the blue bucket and poor it into the red bucket. Then you take one cup from the red bucket and poor it into the blue bucket. Which bucket has the highest ratio between red and blue? Prove it mathematically.

Kind of a trick question. What if the buckets only contained 1.1 cups of paint?

Since the paint is conserved we can say that if after the pouring back and fourth over N cycles (here N=1) an ammount a of paint is transfered from the red to the blue bucket then we can see that the relative ratios are:

[(X-A) Red / A Blue] in the red bucket and [A/(X-A)] in the blue bucket.

As logn as (X-A) is greater than A we have proven the above statement.

Closed room/box with three light bulbs and three switches

There is a room with a door (closed) and three light bulbs. Outside the room there are three switches, connected to the bulbs. You may manipulate the switches as you wish, but once you open the door you can’t change them. Identify each switch with its bulb.


  1. Assuming the lights are not the newer LED bulbs that dont get warm to the touch we can assume if we leave thei light on it will get warm.
  2. We are powering up the box / it does not exsist before time T0. If not it is possible that the heating and cooling times will make the following simplification not work.

Two bulb Solution

{S1 S2} = {a,b} –> Starting test state – wait for an hour.

{S1 !S2} = {a,!b} –> Toggle switch states and open the door.

  1. Light that was constantly on should now be warm and on.
  2. Light that was constantly off should still be off and cold.
  3. Light that was on and is now off should be off and cold.
  4. Light that was off and is now on shoudl be on and cold.

Tests 1 and 2 uniquely identify S1, and you get S2 by process of elimination.

Three bulb Solution

  1. {S1  S2  S3} = {a,  b,   c} –> Starting test state – wait for X mins.
  2. {S1  S2 !S3} = {a,  b, !c} –> Toggle switch and wait for Y mins.
  3. {S1 !S2  S3} = {a,!b, c} –> Toggle switch and pen the door.

Decoding then is

  1. S1. Light that was constantly on should either be ambient temperature and off or very hot and on.
  2. S2. Should either be heated for X+Y mins and off, or ambient and on.
  3. S3. Should either be heated for X-Y mins and on, or heated Y mins and off.

Four temperatures possible: Ambient, Y, X-Y, X+Y

  1. Ambient and off light is S1.
  2. Ambient and on light is S2.
  3. Heated (X+Y) and on is S1.
  4. heated (X+Y) and off it is S2.
  5. Heated X-Y and on is S3 and heated Y and off is S3. So the medium temperature (either Y or X-Y)  is S3.

From 5 above we can collapse the temperature range to Ambient , {Y, X-Y}, X+Y. So just picking a X/Y ratio that allows for adaquade difference in heating and cooling times is all that is needed.

Cut a gold bar into seven pieces with two cuts.

This is how the problem was phrased when I found it on the web.

You’ve got someone working for you for seven days and a gold bar to pay them. The gold bar is segmented into seven connected pieces. You must give them a piece of gold at the end of every day. If you are only allowed to make two breaks in the gold bar, how do you pay your worker?

Overlapping clock hands

Imagine an analog clock set to 12 o’clock. Note that the hour and minute hands overlap. How many times each day do both the hour and minute hands overlap? How would you determine the exact times of the day that this occurs? (Got this problem off of but it was one of their Microsoft questions).