# Mathematical Pattern Puzzle: Fill in the Missing Numbers

## NUMBER PATTERN PUZZLE

Here is an exercise in pattern recognition.

It’s not a linear pattern.

This is an array, so there is a slight geometric element to the pattern.

See if you can figure out the missing numbers in the above puzzle.

Study the four arrays.

See if you can recognize the pattern.

Once you identify the pattern, apply it to the fifth array.

If you scroll down too far…

You will run into the answer.

So stop scrolling down…

If you would like more time to solve the puzzle.

Here it comes.

Now the solution.

Begin with the top left number.

Double the top left number. That makes the top right number. 5 doubled = 10.

Now multiply the top two numbers. That makes the bottom left number. 5 times 10 = 50.

Now add the bottom left number to the top right number. That makes the bottom right number. 50 plus 10 equals 60.

## CHRIS MCMULLEN, PH.D.

• 300+ Mathematical Pattern Puzzles
• Basic Linear Graphing Skills Practice Workbook
• Systems of Equations: Simultaneous, Substitution, Cramer’s Rule

Improve Your Math Fluency. Build fluency in:

• arithmetic
• long division
• fractions
• algebra
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# Five Math Puzzles (pattern recognition): Can You Solve Them?

## MATH PUZZLES

Here is a math puzzle challenge.

Hint: Each of the 5 patterns below has something in common.

Directions: See if you can figure out which numbers go in the blanks.

• 1, 2, 4, 6, 10, 12, 16, 18, 22, _, _
• 4, 6, 10, 14, 22, 26, 34, 38, _, _
• 3, 7, 13, 19, 29, 37, _, _
• 4, 9, 25, 49, 121, 169, 289, _, _
• 5, 8, 12, 18, 24, 30, 36, 42, 52, 60, 68, _, _

If you need help, you can find hints below.

But don’t scroll too far or you’ll run into the answers and explanations.

## PUZZLE HINT

Each pattern above has something in common.

They all involve prime numbers.

A prime number is only evenly divisible by two integers: 1 and itself.

For example, 7 is a prime number because the only integers that can multiply together to make 7 are 1 and 7.

In contrast, 6 isn’t a prime number because 2 x 3 = 6 (in addition to 1 x 6).

Here are the first several prime numbers.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37

Each of the puzzles above relates to these prime numbers.

Here are the answers and explanations to the math puzzles:

• 28, 30. Explanation: Subtract 1 from each prime number: 2 – 1 = 1, 3 – 1 = 2, 5 – 1 = 4, 7 – 1 = 6, 11 – 1 = 10, etc.
• 46, 58. Explanation: Double each prime number: 2 x 2 = 4, 3 x 2 = 6, 5 x 2 = 10, 7 x 2 = 14, 11 x 2 = 22, etc.
• 43, 53. Explanation: Every other prime number: 3 (skip 5) 7 (skip 11) 13 (skip 17) 19 (skip 23) 29 etc.
• 361, 529. Explanation: Square each prime number: 2² = 4, 3² = 9, 5² = 25, 7² = 49, 11² = 121, etc.
• 78, 84. Explanation: Add consecutive prime numbers together: 2 + 3 = 5, 3 + 5 = 8, 5 + 8 = 13, 8 + 13 = 21, 13 + 21 = 34, etc.

## WANT MORE MATH PUZZLES?

One way is to follow my blog. I will post occasional math puzzles in the future.

Another way is to check out my newest book, 300+ Mathematical Pattern Puzzles.

It starts out easy and the level of challenge grows progressively so that puzzlers of all abilities can find many puzzles to enjoy.

A wide variety of topics are covered, including:

• visual patterns
• arithmetic
• repeating patterns
• Roman numerals
• Fibonacci sequence
• prime numbers
• arrays
• analogies
• and much more

The cover was designed by Melissa Stevens at www.theillustratedauthor.net.

## CHRIS MCMULLEN, PH.D.

• 300+ Mathematical Pattern Puzzles
• Basic Linear Graphing Skills Practice Workbook
• Systems of Equations: Simultaneous, Substitution, Cramer’s Rule

Improve Your Math Fluency. Build fluency in:

• arithmetic
• long division
• fractions
• algebra
• trigonometry
• graphing

# Fibonacci Sequence & a Cool Pattern

Image from ShutterStock.

## FIBONACCI SEQUENCE

The Fibonacci sequence adds consecutive terms:

• 1
• 1
• 1 + 1 = 2
• 2 + 1 = 3
• 3 + 2 = 5
• 5 + 3 = 8
• 8 + 5 = 13
• 13 + 8 = 21
• 21 + 13 = 34
• 34 + 21 = 55
• 55 + 34 = 89

Since the last two terms were 55 and 89, we would add these together to get 89 + 55 = 144.

Then you would add 144 and 89 to make 233, and so on.

I saw a cool pattern involving the Fibonacci sequence recently at the Mathemagical Site:

Fibonacci Triples via Mathemagical Site

This involves Fibonacci triples.

A Fibonacci triple consists of three consecutive numbers from the Fibonacci sequence, such as:

• 1, 1, 2
• 1, 2, 3
• 2, 3, 5
• 3, 5, 8
• 5, 8, 13
• 8, 13, 21
• 13, 21, 34

As shown on the Mathemagical Site, the square of the middle number is always one less or one more than the product of the first and third numbers:

Here are a few examples:

• (2, 3, 5): 3 x 3 = 2 x 5 – 1
• (3, 5, 8): 5 x 5 = 3 x 8 + 1
• (5, 8, 13): 8 x 8 = 5 x 13 – 1

Curious about this, I’ve been working through the algebra, and finally came up with an algebraic proof, which follows.

My proof is algebraic and not necessarily obvious, but since I worked it out, I thought I would share it here. 🙂

We begin with two facts about the Fibonacci sequence:

These are two different ways of saying that if you add two consecutive numbers from the Fibonacci sequence, you get the next number in the sequence.

Now solve for xn in each sequence:

Multiply these together:

Foil this out:

Recall that

Plug this into the first term on the right-hand side of the previous equation:

Distribute:

Two of these terms cancel (the remaining terms are rearranged):

Believe it or not, this basically concludes the proof. The remainder is basically interpreting this result.

This is a recursion relation that relates the square of the n-th term to the square of the previous term (xn-1 times itself).

Following is the Fibonacci sequence, labeling values of n:

• n = 1 is 1.
• n = 2 is 1.
• n = 3 is 2.
• n = 4 is 3.
• n = 5 is 5.
• n = 6 is 8.
• n = 7 is 13.
• n = 8 is 21.

Let’s plug in n = 3 and see what happens:

If instead you plug in n = 4, you get:

Now just plug in these two expressions (x3x3 – x4x2 and x4x4 – x5x3) into the previous recursion relation and you can prove that all of the Fibonacci triples satisfy one of the following relations:

That is, if x3x3 – x4x2 = 1 and x4x4 – x5x3 = -1, the previous recursion relation gives similar expressions for x5x5 – x6x4, x6x6 – x7x5, and so on.

## CHRIS MCMULLEN, PH.D.

• Algebra Essentials Practice Workbook with Answers
• Trigonometry Essentials Practice Workbook with Answers
• Learn or Review Trigonometry: Essential Skills

# Tessellations Hexagon Square Triangle Rhombus Trapezoid Star Patterns

## TESSELLATIONS

A tessellation is a repeated two-dimensional geometric pattern, with tiles arranged together without any space or overlap.

Simple tessellations can be made by creating a two-dimensional lattice out of regular geometric shapes, like equilateral triangles, squares, and hexagons. Not any regular polygon will work, however. For example, it won’t work with pentagons.

Tessellations can also be made from irregular polygons. (A regular polygon is one with equal sides and angles.) All quadrilaterals can form tessellations. (Quadrilaterals are polygons with four sides.) Although regular pentagons don’t tessellate, some irregular polygons can (such as the pentagon made by placing an isosceles triangles on a square, as children often do to draw a simple picture of a house).

There are many other shapes that tessellate, such as stars combined with other shapes. Even arrangements of curved objects can tessellate. Some of the more extreme examples of this can be seen in M.C. Escher’s artwork.

The lattice structure below can be shaded in several different ways to create simple geometric patterns that tessellate:

For example, here is a tessellation composed of hexagons:

Here is another made from triangles:

This one is made with squares:

The same pattern can make a tessellation with stars and hexagons:

Here are a variety of basic geometric shapes that can tessellate from this same pattern, including a hexagon, triangle, square, trapezoid, parallelogram, pentagon (irregular), rhombus (diamond), and rectangle: