Gamma wave meditation aid

I built a circuit that flashes a 6 by 6 array of LED lights at
40 Hz (40 flashes per second).  This 40 Hz frequency
stimulates the brain to strengthen the level gamma waves
that it normally produces at that frequency.  This is said to
have a calming effect on your mood and to generate a sense
of well-being.  And is said to be an aid to meditation.  Also,
a 40 Hz flashing LED array, might alleviate the symptoms of
Alzheimer's disease, a small study has suggested.
Here is a link to the circuit:

The LED array I use in the circuit is this product on Amazon:

You could power it with a power bank, external battery for
charging smartphones. There are many of these on Amazon.
Also you could power it with a 5 volt voltage-regulated wall wart.


About Butterscotch and How to Make Butterscotch Flavor Powder

About Butterscotch
      My sources for this are many Google searches about butterscotch on the internet. 
First, there is no such thing as "true butterscotch".  The many recipes I found have various
ratios of sugar to butter, and there is a variation in the ingredients.  So there is only
"version A" of butterscotch, "version B", "version C" and so on.  The commonality
among most of them is that they contain sugar, butter, salt, and molasses.  The sugar
can be light brown sugar, dark brown sugar, or just plain sugar.  If it's just plain sugar,
molasses is added to the recipe; otherwise, molasses is optional.  If molasses is added
it can be plain molasses or, occasionally, blackstrap molasses.  The sugar to butter
ratio varies greatly — anywhere from 4:3 to 16:1 by weight, I've read.

The sugar, butter, salt, and molasses are mixed together and cooked until the temperature
is raised to anywhere between 245 degrees F to 310 degrees F. 

One question I asked myself is why, if molasses is just mostly caramelized sugar (as many
sources on the internet say), why, if it's added separately, is it always cooked along with the
other ingredients and not added later?  After further research on the internet, I found the
answer: Molasses is NOT caramelized sugar!  Why?  To refine sugar, sugar cane is boiled in
a vacuum at 160 degrees F.  160 degrees F is well below the caramelization temperature of
sugar, which begins at about 320 degrees F.  Also at 160 degrees F the Maillard reaction
(which I'll get to shortly) is very slow, so molasses has undergone very little Maillard reaction.

Caramelization is not important for butterscotch because butterscotch is not caramelized, but
the Maillard reaction is very important since it is what develops the butterscotch flavor.  The
Maillard reaction takes place at an optimal rate between 270 and 310 degrees F.  To avoid

caramelization from occurring, among all but one of the recipes  I've seen for butterscotch,
the maximum cooking temperature is 300 degees F.

How to Make Butterscotch Flavor Powder
      I like the flavor of butterscotch a lot, and after some experimenting I figured out how to
make a butterscotch flavor powder that I could add to other things, such as yogurt.  The
following recipe is based on recipes for butterscotch I've found on the internet, plus a
couple of my own ideas, which seem to be original.

60 grams      dark brown sugar

30 grams      unsalted butter
1/8 tsp          salt (not sea salt)

1 1/2 Tbsp    blackstrap molasses (optional, to make the flavor "extra butterscotchy")
2 tsp             water (approximarely)

      Mix the ingredients together in a pan, adding just enough water to wet down the
other ingredients, set a stovetop burner in the medium heat range, and, stirring 
constantly, cook slowly until the temperature of the mixture is a carefully measured
310 to 315 degrees F.  Cooking slowly is important for giving the Maillard reaction time
to take place. (The Maillard reaction is what develops the butterscotch flavor.)  For
measuring the temperature, I use an infrared surface temperature thermometer. 
There are many brands of these infrared thermometers on Amazon.

When the mixture is cooked, pour it immediately (not letting it harden) onto a flat
surface that has been covered with a sheet of parchment paper, and let the mixture
cool for, say, 20 minutes, until it is rock solid.  Then crack it up with your food-gloved
hands, transfer it to a blade type coffee grinder, and grind it into a fine powder.  This
may require several "transfer-grind" operations.  Store the powder in a refrigerator,
and use it as desired.  When using the powder, also add some vanilla flavor or extract.

Serving size 1 Tbsp.  Makes about 7 servings.

Minus Times Minus Equals Plus

Here’s an example I like of a negative number times another
negative number equaling a positive number:

I have been giving away five dollars each minute.
Currently, at time t = 0, I have zero dollars, but I am continuing
to give away five dollars each minute in the form of IOU’s.
Thus, the equation for the number of dollars I have at time t is
D = -5t, where D represents dollars.

Using that equation, calculate how many dollars I had four
minutes ago.

Vitamins for the Mind

                               The 4:00 Meeting
Tom and Bill are standing at two places on a straight road.  Tom starts
walking toward Bill and arrives at Bill’s original place 11 minutes after
Bill had left.  At perhaps a different time, Bill starts walking toward Tom
and arrives at Tom’s original place 15 minutes after Tom had left.  When
each reaches the other’s original place, he immediately turns and starts
back, and they meet in the center at 4:00.  Assuming that they walked at
constant rates, when did each start to travel on the road?

                          Alice Forgets Her Purse
On her way out of Macys, Alice walks down a down-moving escalator
in 50 steps.  When she reaches the bottom, she suddenly remembers
she forgot her purse, and she turns and runs back up the escalator in
125 steps, stepping five times as fast as she went down.  How many steps
are on the surface of the escalator?

                               The Twelve Coins
You have 12 coins.  They are identical, except that one of them is either
heavier or lighter than the rest.  In three weighings on a balance scale, find
the odd coin and whether it’s heavier or lighter.

Complete Protein Organic Whole Grain Bread Recipe

It took me many trials and many adjustments to develop this recipe for a nicely rising whole grain bread.  Note: The amounts may look over obsessive in their precision, but they are just the amounts I somehow ended up with over many trial loaves.  They probably don't need to be all that exact.

Dry Ingredients                                                    Liquid Ingredients
78g  King Arthur organic whole wheat flour         322g   water
        (not stone ground)                                        29g     clover honey (or other mild
79g  organic whole amaranth flour                       1/2 Tbsp  extra virgin organic olive oil,
        (not stone ground)                                                       first cold press
79g  organic whole quinoa flour (not stone ground)
79g  organic whole buckwheat flour (not stone ground)
70g  Bob's Red Mill vital wheat gluten flour
23g  King Arthur Baker's Special Dry Milk (a nonfat non-instant dry milk)
7.7g salt (not sea salt)
6.1g Fleischmann's bread machine yeast

1.  The wheat, amaranth, quinoa, and buckwheat should add to 315g — measure these without rezeroing the scale: 78g, 157g, 236g, and finally 315g.

2.  For baking the bread: In my Zojirushi BBCC-X20 bread machine I used the Home Made baking course in Memory 1, which I set to the following numbers of minutes: Preheat 30, Knead 23, and Rise1 45.  Then I  transferred the loaf to a 8 1/2" x 4 1/2" loaf pan and did a final rise in the bread machine for about 44 minutes (a "Rise2" so the surrounding temperature would be 82.4 degrees F).

3.  Put the loaf into an oven preheated to 425 degrees F and immediately lower the temperature to 400 degrees F. After 9 minutes lower the temperature to 350 degrees F.  Tent the loaf after 19 minutes and continue baking until the internal temperature of the loaf is 201 degrees F (94 degrees C).  The total baking time should be about 34 minutes.

4.  Makes about a 1 1/2 pound loaf.

Free Will

I have been reading Sam Harris's latest book, "Free Will", in which he
claims that free will does not exist.  Basically, he refers to some scientific
studies that show that all actions that you consciously decide to take
have actually been decided for you, totally unconsciously, in your
unconscious mind a few hundred milliseconds, or sometimes a few
seconds, before it enters into your conscious mind and you "decide" to
take it.  But since your unconscious mind has already decided, your
conscious decision is an illusion — you actually had no free will about it.

Harris is aware of the idea of quantum uncertainty, and acknowledges
that Martin Heisenberg, a biologist, has found that certain processes in
the brain occur quantum randomly (thus truly randomly), but says that
has nothing to do with free will.

But I think it may.  Consider this example, which is based on an actual
experience I had in solving a problem:  I had previously noticed that
when you heat a UPS label on a package with a hair dryer and peel it
off, it leaves a sticky area where the label was.  My problem was to
figure out how to get rid of the stickiness.

I don't know much about how the brain works, but as far as I know, there
is nothing to contradict the hypothetical mechanism I will suggest.  And
it coincides with my thinking process in solving the problem:  When you
are trying to solve a problem, perhaps there is an unconscious
mechanism in the brain that generates possibilities at random,
randomly cranking through and randomly associating past events,
objects, tools, materials, devices, et cetera that you know about.  And
when it finds an association that might relate to your problem, presents
it to your conscious mind for further consideration.  In my case of how
to get rid of the stickiness here's how it might work:

My thinking process:
How to get rid of the stickiness???  — think… think… think… (unconscious
mind cranking out a possibility).  Conscious mind:  Aha!  Try rubbing it with
a white pencil eraser.  Hmm — no, that sounds like it would just rub the
sticky around, but not get rid of it.

So,   — think… think… think…
(unconscious mind generating another random possibility).  Conscious
mind:  Aha!  Try spreading glue over the stickiness and letting it dry. 
Hmm — Maybe.  I have some Elmer's white school glue, I'll try that. …
No — sorry — it didn't work.

Back to   — think… think… think…
(unconscious mind generating yet another random possibility). 
Conscious mind:  Aha!  Try rubbing over it with a wax candle.  Hmm —
interesting.  Strange, but who knows?  So try it. … Yes!  Amazing.  It
worked perfectly!  Problem solved.

That's a pretty good description of how my thinking went in my
conscious mind.  Of course, I have no idea what actually went on in my
unconscious mind.  But where do these ideas that just pop into your
head come from?  Especially the one about the wax candle.  The few
candles I even own have been sitting on a back shelf for several years,
untouched.  I haven't even the vaguest (conscious) clue about where
that idea came from.

But, if there is a part of your unconscious mind that randomly
generates ideas for your conscious consideration and decision as to
whether or not to act on, I think that would be an example of free will.

Derivation of the Formula for Mortgage Payments

Derivation of the Formula for Mortgage Payments

Copyright (C) April, 2012 by Bob Day. All rights reserved.

A while ago, I wanted to know the reasoning behind the amount of my payments on my home mortgage. I looked in some accounting books, and all they gave was the formula. None of them gave the rationale behind it. So I sat down and derived it myself. It's not too hard.

Say we borrow an amount "A" at an interest rate of "r" per payment period. (If the payments are made monthly, "r" is the annual interest rate quoted by the bank divided by 12.) We pay back the loan in "N" payments or periods.

For example, for a 30 year mortgage on which payments are made monthly, N would be 30×12 or 360. After N payments, each of the amount "P", the loan is paid off and the amount we owe is reduced to zero.

Derivation of the Formula
The Initial amount we owe is A. At the end of the first payment period, the amount we owe has increased by rA, one payment period of interest, and we make a payment, P. So the total amount we owe after one period is
: A + rA – P, or A(1 + r) – P. We note that this A(1 + r) – P is not only an amount, but also an operator; that is, given the amount of principal outstanding at the beginning of any period, we can apply it to determine the amount of principal remaining at the end of the period.

So applying the operator A(1 + r) – P to the amount A(1 + r) – P remaining at the end of the first period, we get (A(1 + r) – P)(1 + r) – P as the amount remaining at the end of the second period. Similarly, at the end of the third period, the amount of principal remaining is: ((A(1 + r) – P)(1 + r) – P)(1 + r) – P. After N periods (applying the operator and then expanding), the amount remaining will be:

A(1 + r)^N – P( (1 + r)^(N-1) + (1 + r)^(N-2) + (1 + r)^(N-3) + … + 1 ) = 0  [Equation 1].

It equals zero, because after N periods the loan is paid off. Considering just the (1 + r)^(N-1) + (1 + r)^(N-2) + (1 + r)^(N-3) + … + 1 portion, we can reverse the order of its terms and rewrite it as: 1 + (1 + r) + (1 + r)^2 + … + (1 + r)^(N-1)

Representing this series by "S", and letting "R" equal (1 + r), we get:
S = 1 + R + R^2 + … + R^(N-1)
So, RS = R + R^2 + … + R^(N-1) + R^N
Subtracting: S – RS = 1 – R^N, and so S = (1 – R^N) / (1 – R)
Inserting this value for S back into Equation 1:

A(1 + r)^N – P( (1 – R^N) / (1 – R) ) = 0

Finally, Solving for P, the amount we pay each period, we get:

P = rA / (1 – (1 + r)^(-N))

I checked this formula against my own mortgage amount and payments and it agreed exactly! Voila! For example, for a 30 year mortgage for $300,000 at an interest rate of 6% per year paid monthly, the parameters are: A = 300000 (the mortgage amount) r = 0.06 / 12 = 0.005 (the monthly interest rate) N = 30 x 12 = 360 (the number of payments) And the monthly payments would be: 0.005 x 300000/(1 – 1.005^(-360)) = 1798.65 dollars per month.

Another Way: Approximation with a Differential Equation
We can also use a differential equation to get a very close approximation of the payments on a mortgage.  A while ago I was trying to figure out how long it would take a bug walking along a stretching rubber band to get to the end.  After I solved that problem, it occurred to me that the problem of mortgage payments could be solved in a similar way.  It's a nice example of how a differential way of thinking can be used to solve a real-world problem.  Perhaps many problems in finance and economics can be solved using a differential approach.

We start by looking at how fast the mortgage is being paid off:  We now let A(t) be the amount we owe on the mortgage as a function of time.  Note that A(0) is the amount of the mortgage, the amount we borrowed.  Each month, the bank adds an interest amount of r * A to the mortgage and we make a payment of P.  Consequently, the amount we have remaining to pay on the mortgage changes by: dA/dt = r * A – P each month. 

To solve this equation for A requires a little bit of mathematical gymnastics, but it's strictly cookbook.  It can be very easily solved by entering "solve (dA/dt = r * A – P)", without the quotes, into WolframAlpha at and clicking on the = sign. 

The solution is: A(t) = P/r + C e^(rt), where C is a constant we need to evaluate. After a time T, the mortgage will be paid off, so we have: A(T) = P/r + C e^(rT) = 0. Solving for C, we get, C = -P/r e^(-rT). Replacing C in the solution, A(t) = (P/r) (1 – e^(r (t-T)). So, A(0), the amount of the mortgage (the amount we borrowed) is: A(0) = P/r (1 – e^(-rT)). And finally, solving for P we get: P = r A(0) / (1 – e^(-rT)). For A(0) = 300000 dollars, i = 0.06 / 12 = 0.005 percent per month, and T = 360 months, we get: P = 1797.05 dollars per month, very close to the amount we calculated before.  (But, of course, not quite good enough for the bank!)

How to Lose Weight: Calories Aren’t Everything

… Calories are the only thing.  If you burn more calories than you take in you will lose weight — beginning, middle, and end of story.  Maintaining a negative calorie balance is the only thing that matters for weight loss.

The therapy professions and pharmaceutical companies would have you believe that obesity is an uncontrollable disease or an addiction.  They want you to pay for therapy or buy their drug.  Both are total nonsense.

But, losing weight is difficult.  It's a long, lonely, and solitary road.  It's not just a matter of going on a diet for a while, and it requires changing much more than just your diet.  It requires a permanent change in lifestyle, including discarding friends who disparage or try to discourage your efforts.  You, yourself, inside your soul, have to decide that losing weight is something you want to do.  Not because friends said you should lose some weight; not because your doctor is on your case about it.  You look at yourself in the mirror and decide you don't like the way you are and decide that you want to change your life.

I know all this for a fact.  I've lost over 50 pounds, gone from being overweight to being a few pounds above underweight, and have kept it off for more than ten years.

1)  Diet Programs like Weight Watchers can help you to find a weight losing diet, but not for support.  If you need support from others, you'll probably fail.  Your desire to lose weight, which requires establishing a new, permanent, lifestyle, must come from within.

2)  Gaining Weight as you get older you're older is *not* inevitable. It's just another excuse.

3)  Exercise has many health benefits, but it is not necessary for weight loss.  It can help only if you don't use exercise as an excuse to eat more.


1)  Eat mostly to get the nutrition your body needs, and less for enjoyment.  Establish a healthful diet and learn to enjoy healthful foods, and make eating less a part of your life.

2)  Three meals a day: breakfast, lunch, and dinner.  That's it.  No snacks, and no "in between" meals.

3)  Give up sugar.  No sugar in coffee, soda, or on cereal.  Give up fruit juice — it's mainly just another form of sugar.  Water is the only liquid you need.

4)  No alcohol.  Alcohol has no food value, alcohol is just empty calories.

5)  In the beginning, establish a very regulated moderate calorie diet.  Don't follow any sort of fad.  Just pick a selection of foods that add up to a normal balanced diet  — whole grains, veggies, fruit, dairy, a little meat, etc.  But start out by having exactly the same three meals each day — the same foods and the same amounts.  Weigh the portions on a scale.  Consider frozen dinners.  Healthy Choice, Lean Cuisine, Kashi, Smart Ones, and probably other brands have several that are low in calories (350 or less) and saturated fat, 25% daily value or less of sodium, and high in fiber — no need to weigh these.

6)  Round out your diet with supplements for nutrients that your diet does not contain enough of.

7)  Weigh yourself every day on a 0.2 lb. accuracy scale.  Your weight will fluctuate, but with a constant diet it should trend down over every two or three days.  If it doesn't, eliminate items from your diet or reduce the size of portions until your weight does go down.  (If you don't have a 0.2 lb. accuracy scale, I'd recommend the EatSmart Precision Plus Digital Scale, which is sold on Amazon.)  Don't obsess over the scale — let it be your friend and point the way to a weight losing diet.

8)  When you have achieved a weight losing diet, then you can start making adjustments to add variety, but make sure that you keep losing weight.

9)  Establish a routine of regular daily exercise.

10)  When you're down to the weight you want to be, you can adjust this diet to be a weight maintaining diet.  Make changes for variety, but keep to the approach to eating recommended here.  That can be your permanent diet for life.  Eat mainly to get the nutrition your body needs and less for pleasure  — find pleasure in other things.