# Category Archives: mathematics

## Let’s have fun with your Phone Number

Want to pull off a great trick that will make you the toast of the cocktail party circuit?  A trick so fiendishly clever that men will envy you, and women want to flirt with  you?  Well, this probably isn’t that trick.  But it’s pretty good, and it involves the concept of numerical congruence dreamed up by a famous German mathematician named Carl Friedrich Gauss.  Now, I know you.. you want me to get right into it, you little scamp, so here it is:

A) Write down your phone number on a piece of paper.  Area Code too.

B) Using those same digits in your phone number, randomize that phone number any which way you want, to create a Dummy Number.  Use cards, or (like me) put slips of paper with the numbers on them in a cup and draw them out in a row for dramatic effect.. reciting pertinent facts from the life of Carl Friedrich Gauss.

C) Subtract the smaller number from the larger number.   It doesn’t matter if the real phone number (yours, presumably) is larger or smaller than the Dummy Number.

D) Add up all the digits created by the difference between the larger phone number and the smaller phone number.  This will likely create a two digit number.

To give you a handy visual, I’ve encapsulated steps A-D in this handy graphic lifted from Excel, and using the made up phone number of 212-555-2379.  I randomized them with pieces of paper being pulled out of a salad bowl.  See the highlighted line for the Dummy Number.  Important!  use YOUR OWN number, not this made up number.

Observe THIS table of secret cannibalistic symbols, taken from ancient scrolls of forbidden wisdom:

Starting with the Pentagram up top (As number one) move clockwise, with the triangle as number two, etc. Go around the circle as many times as your number from Step D.

What symbol did you land on?  I have a guess, here it is.  What’s more, here’s the trick: It will land on that symbol EVERY, SINGLE TIME.  Every one!

Now, isn’t that handy?  Men will mutter and women will swoon.  At the very least, you’ll get a free drink out of it if you play your cards right.

Psssst.. it’s a mathematics thing, not magic, but you know that already. To understand it, you might have to read up on mathematical congruence.  If two numbers have the same remainder when divided by a number k, they are congruent to the number k.  The number K is called a “modulus”.  Example– 16 and 23 buth have a remainder of 2 when divided by 7 and therefore are congruent modulo 7.  Since 9 is the largest digit in the decimal number system, the sum of the digits of any number will ALWAYS be congruent modulo 9 to the original number.  Note that there are 9 digits in a telephone number.  Scrambling the digits can’t change the digital root of a number, so basically you end up with multiples of 9 + the digital root subtracting multiples of 9 + the same digital root, equaling a multiple of 9 +0.  For our “cabalistic symbol counting” purposes, it will always give you one result.  Nifty, eh?  Remember that cool “mind reading” t-shirt from a previous post?  It works on exactly the same principle, just different numbers.

Let’s do a mind-meld.  See my shirt?

Yeah, I know, I’m a real great model.  Dead sexy!

Imagine you’re a tiny, doll-sized person. Standing on the green circle with an X on it, imagine a number between 5 and 20. Starting with the FIRST BLANK GREEN CIRCLE, Walk that number of circles around the circle the same number of circles as your number. Stop. Reverse. Walk that same number of circles back, staying in the circle of white symbols until you hit your number.

Next, think real hard about your symbol, and Email or Comment me with a single word describing it.. SQUARE, MOON, CIRCLE, etc. Here’s the thing.  I’ve already guessed your answer and I’ll visualize my response back to you with an image– circle, square, triangle etc. Tell me if I’m right or not.  You may have to email me to not spoil for the next person.

Or, what the heck, just check here.  Was I correct?

Please, no snarky comments.  Sure, it’s a trick, and a danged fine one.  Thank you, Richard Wiseman.

## April 2014 Domino Number Pair Puzzle

Dominoes. See the number pairings? 4,4 .5,1. 0,4. 0,2. 5,4 (front), etc.

You know, it’s been a regular month of Sundays since I puzzled a puzzle to this blog, and I used to do that regularly. So here we go! For APRIL 2014, we present the Domino Number Pair Puzzle.  Think of this grid of numbers as DOMINOES (pairs of numbers).  Where would you draw boxes around the number grid to match the pairings in this grid of numbers:

April 2014 Domino Number Puzzle: Questions

Here are the possible combinations:

0,0
0,1 1,1
0,2 1,2 2,2
0,3 1,3 2,3 3,3
0,4 1,4 2,4 3,4 4,4
0,5 1,5 2,5 3,5 4,5 5,5
0,6 1,6 2,6 3,6 4,6 5,6 6,6

You don’t have to answer. It’s a thought exercise. Note that 4,6 is already filled in on the top left corner.

## 23 May 13 Puzzler: The Domino Problem

Today’s little mental hotfoot comes from GIFTED MATHEMATICS, a great website that helps make mathematics more approachable.

Prep: Take one full set of dominoes and remove the tiles with blanks, leaving you with 21 tiles.

The diagram below shows 8 rectangles placed at the vertices of an octagonal wheel. Each rectangle is to have one domino tile placed inside it. Every tile consists of two squares, each with some pips inside it. The number in the centre of the diagram is the product of the 4 squares taken from the 2 dominoes that are diametrically opposite each other. All four such products must equal the same total and be formed from different tiles from your set of dominoes. Next, orient each tile so that the square with the smallest number of pips is closest to the centre of the diagram, thereby forming a kind of “inner circle”. Calculate the sum of these eight squares, the “inner sum”.

The aim is to find the largest possible “inner sum” of the lowest value squares of the 8 dominoes placed so that the product of the squares of diametrically opposite dominoes is constant.

A Domino Wheel: Upper Secondary Mathematics Competition Question

As an example, the diagram shows two domino tiles placed diametrically opposite each other. The product of their 4 squares is 5x2x3x4 = 120. The partial “inner sum” is 2+3 = 5. If this were a solution, the aim would be to complete the diagram with 6 more dominoes whose products (as defined above) are all 120, then to add the squares in the “inner circle”. Finally, you would need to establish that your final answer is, indeed, the maximum possible.

This would make a neat little game, too.

## Jan 13: Connect Sums Puzzle

It’s the January 13 Connect sums puzzle! Wheeee!

Here you go, I’ve been posting far too many of these over at Facebook when I used to post them here on WordPress. Here’s a connect sums game. Can you connect the numbers into four (4) groups by drawing horizontal and vertical (but not diagonal) lines between the number circles? The four groups must all sum to the number 18. Each circle may only belong to one group. Lines may not cross each other. The first line is drawn in the top right column. Since It would be difficult to express your answer in a picture on WordPress, you can express your answer in brackets, like so: group 1: green 6 down to red 2. purple 4 left to green 6, etc…

Any schoolkid taking more than algebra 1 will probably tell you that this form:

is the algebraic expression of a Quadratic Equation.  Quadratic equations are important for a lot of reasons, but the most real-life example is measuring land ownership, which has been with us since civilization existed.  The Latin word for “square” (a handy form for measuring land) is quadratus, which is where “quadratic” comes from.

Solving for x is probably something most post-algebraic people can do with a little boning up on theory:

That’s modern notation.  Naturally.  You can find this in a textbook, or wikipedia, or pretty much anywhere.  However, most people don’t think like algebraic notation.  I know I don’t.  The Babylonians certainly didn’t.  What’s cool– blog-postingly cool… is that they had a rule of thumb method for handling quadratics which pretty much did the same thing as algebraic notation.

This contains the solution to the following problem:

The area of a square, added to the side of the square comes to 0.75.  What is the side of the square?

The tablet actually works out a solution step by step, using the Babylonian rule of thumb for quadratics.  Like so:

1. “I have added the area and the side of my square. 0.75”
2. “You write down 1, the coefficient”
3. “You break half of 1. 0.5”
4. “You multiply 0.5 and 0.5.  0.25.”
5. “You add 0.25 and 0.75.  1”
6. “This is the square of 1”
7. “Subtract 0.5, which you multiplied”
8. “0.5 is the side of the Square”

That’s pretty impressive.  Using the mathematics they had on hand, and logic, they figured out something pretty danged important.  This is roughly arriving at this formula:

to solve