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Archive for the 'Solutions' Category

Answer One: To Laugh or to Cry?
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Answer Two: Deep Sleep
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Answer Three: Food Court
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Answer Four: Phone For Help
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Problem Five: How Far Can You Go?
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Answer One: Moving Troubles

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Answer Two: Legos for All

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Answer Three: How many coins?

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Answer Four: Mmm… Sugar Cubes

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Answer Five: Lucky Number 13

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Answer Six: an eighth is greater than a quarter?

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Answers to Problems on Oct 27th

Answer One: Spot the Differences VIII

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Answer Two: Sudoku VI

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Answer Three: Coin in the Bottle

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Answer Four: All Five Vowels

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Answer Five: Detention Essay

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Answer Six: Fifty

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Answer One: The Blood of the Earth
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Answer Two: The Scrapbookers
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Answer Three: Royalty?
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Answer Four: Self-Esteem Builder
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Answer Five: Speeding Blues
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Answer Six: Tinman’s Pick
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Answer Seven: Screaming and Soaring
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Answer Eight: 1 to 12
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Answer Nine: Big Splash
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Answer Ten: Blessing or Bane
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Solution to EST MODUS IN REBUS

This is going to be quite a long post, since the solution is quite involved. Here is the original problem.

This is the diagram in question:

Est modus In rebus Math Puzzle

First, we note that there are both alphabets and asterisks in the equation. We see that there are 12 different letters in total, so while the same letters always represent the same digit, each digit may be represented by more than one letter. For example, while “E”s may always represent “3″, other letters, such as “M”, may also represent “3″.

To simplify our analysis, let’s call the eight digit multiplier (ESTMODUS) X and the other (INREBUS) Y. Since there are seven partial products (the seven lines of asterisks in the middle), we can deduce that there are no “0″ in Y. We also note that the last two digits of X, Y, and their product are the same, namely “US”. Thus, the square of “US” must also end in “US”; for example, 25^2=625 and 76^2=5776. It turns out that these are the only two possibilities (among two digit numbers, which we are looking for here), and so “US” must be 25 or 76. To prove this point, let the two digit number “US” be z, then the last few digits of the product X \times Y must be z^2. But the last two digits of z^2 is just z, therefore z^2-z=z\left(z-1\right) must be divisible by 100. From this, we can see that either z is divisible by 4 and z-1 is divisible by 25, or z is divisible by 25 and z-1 is divisible by 4. In the first possibility, z=76; in the other possibility, z=25. Q.E.D.

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Answers to Problems on Oct 24th

Answer One: Spot the Differences VII

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Answer Two: Sudoku V

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Answer Three: Missing Square Puzzle

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Answer Four: Crazy Driver

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Answer Five: 1=2

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Answer One: Sound of Honour
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Answer Two: Palmer’s Notation
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Answer Three: Train Problem
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Answer Four: Three Statements
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Answer Five: 911 Call
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Answer Six: Don’t Get Stressed
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Answer Seven: Cars
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Answer Eight: Tribal Truth
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Answer Nine: Colors
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Answer Ten: Contractions
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Answers to Problems on Oct 20th

Answer One: Spot the Differences VI

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Answer Two: Sudoku IV

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Answer Three: Too much i’ th’ sun

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Answer Four: A Knight’s Tour

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Answer Five: A Knight’s Tour II

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Answer Six: Hopscotch

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Answer Seven: 1/7 to 1

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Answer Eight: Ice And Out

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Answer Nine: Bacterial Growth

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Answer Ten: Wrong!

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Answer One: Nefarious Liars

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Answer Two: More Nefarious Liars

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Answer Three: A Prime Cube

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Answer Four: Unfair Professor

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Answer Five: Night at Bar

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Answer Six: Sea shells by the shore

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Answer Seven: Not a Knot

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Answer Eight: Roar…

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Answer Nine: Chess Championship

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Answer Ten: The Passcode

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Answer One: An Ingenious Riddle
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Answer Two: A Simple Question
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Answer Three: Watch Theft
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Answer Four: A Unique Series
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Answer Five: Soccer Project
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Answer Six: What Word
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Answer Seven: Menacing Matchsticks
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Answer Eight: Condemnation
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Answer Nine: Dinner Time
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Answer Ten: Alphabetically Challenged
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Answers to Problems on Oct 17th

Answer One: Spot the Differences V

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Answer Two: Sudoku III

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Answer Three: Silly Sally

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Answer Four: Tough Sequence

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Answer Five: Right and Left

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Answer Six: Consecutively

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Answer Seven: Crossing the River

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Answer Eight: Angry and Hungry

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Answer Nine: Dilemma

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Answer Ten: Another Sequence

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Answer One: Am I a Signal?
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Answer Two: Little Women
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Answer Three: Middle Initial
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Answer Four: Tying the Knot
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Answer Five: Clearly Confused
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Answer Six: The Four Doors
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Answer Seven: Mud Hook
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Answer Eight: Sequential Times
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Answer Nine: Games
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Answer Ten: Heavy Lifting
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Answers to Problems on Oct 8th

Answer One: Dividing Tokens

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Answer Two: Spot the Differences IV

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Answer Three: Sudoku II

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Answer Four: City Luncheons

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Answer Five: Finding Perfect Squares

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Answer One: Find the Differences III

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Answer Two: Smoke Direction

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Answer Three: Sudoku

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Answer Four: The Alphabet Challenge

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Answer Five: Buckets of Water II

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Answer Six: Concerning a Check

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Answer Seven: Fifty-Seven to Nothing

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Answer Eight: Paula

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Answer Nine: Mayhen

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Answer Ten: The Forest

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Answer One: Country Scramble
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Answer Two: Numbers
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Answer Three: Reverse Words
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Answer Four: Twisted Personas
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Answer Five: Four Sisters
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Answer Six: One-armed Bandit
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Answer Seven: Eating For Free?
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Answer Eight: Word Pyramid #1
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Answer Nine: Yellow Bark
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Answer Ten: Operation
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