Math Exercises for the Brain

OLLI

Winter, 2017

Math Exercises for the Brain

For decades, it was thought that the brain inexorably lost nerve

cells, or neurons, as we aged, and that we could never grow new

ones.

We now know that the brain isn’t “hard-wired” as we once

believed. It can grow new neurons, tie them together to form new

neural pathways, and rewire itself. All we have to do is to keep

thinking, learning, experiencing, and remembering – “exercising

our brain”.

What goes on in these different parts of the brain? How will doing math puzzles improve them?

Pattern recognitionThese puzzles work your executive functions in your

frontal lobes by using your pattern recognition,

hypothesis testing, and logic.

Example: What comes next in this sequence?

6, 20, 27, 41, 48, 62, ___

6, 20, 27, 41, 48, 62, 69

Add 14, add 7, add 14, add 7, …

Logic ProblemsThese puzzles touch on analytical functions like abstract

reasoning, hypothesis testing, and implicit assumptions residing in

your frontal lobes, as well as your creativity in finding novel

solutions to problems and emotional memory.

Example: Jan and Jill ran into each other at a reunion. They

hadn’t seen one another since their school days. “Good grief,” Jan

told Jill, “you must have gained a hundred pounds!” Jill was not

overweight. So, why was Jan’s estimate accurate?

However, when it comes to logic problems, you

may have to use your reasoning skills more than

your memory skills, as it is likely that you do not

know the answer immediately. In this case, the

frontal exercise is more intense.

You have five segments of chain and want to connect them

all into one single length of chain.

It costs 5 cents to break a link and 10 cents to weld it again.

What is the least it would cost to join the five segments

shown?

Is your answer 45 cents? Good for you!

It costs 5 cents to break a link and 10 cents to weld it again.

What is the least it would cost to join the five segments

shown?

Break one section into three separate links: 15 cents

Use the “opened” links to connect the other sections: 30 cents

Memory relies mostly on some temporal (in green)

and frontal (in red) areas of the brain. These may

be the areas that will get stimulated when you try

to recall arithmetic facts or how to use them with

different operations.

Do this in your head:

Add one thousand twenty and one thousand twenty…Now add

twenty to your answer…Add twenty again…Now add ten…And

add ten again. What’s your total?

The answer is 2,100. Is that your answer? Magnificent!

Did you come up with 3,000? That’s what most people get.

You need more practice!

Algebra and ComputationsIf you have a good memory for “math facts,” such as simple sums

and differences and the multiplication tables, then you don’t have

to waste mental energy on performing these calculations and can

concentrate instead on more interesting problems.

Example: Use four 2’s and arithmetic operations so that the

resulting answer is “1”.

Use four 2’s and arithmetic operations so that the resulting

answer is “1”.

22

22

2 2

2 2

2.2

2.2

2

2

2

2

The Trachtenberg Speed System of Mathematics was

developed by a Russian Jewish engineer, Jakow

Trachtenberg, in order to keep his mind occupied while being

held in a Nazi concentration camp. The Trachtenberg system

allows one to perform arithmetic computations very quickly.

Examples:

• Quick multiplying by 11

• Multiplying two 2-digit numbers

Multiply: 345,624 11

To do this, put 0’s at the beginning and end. Then, starting at

the right, add the adjacent digits together, adding carry-overs to

the next sum.

0 3 4 5 6 2 4 0

V V V V V V V3 8 0 1 8 6 4

345,624 11 = 3,801,864

Multiply :

45

23

Multiply two right numbers; keep track of carry-over.

Multiply two “criss-cross”, add products plus carry-over.

Multiply the two left numbers (add the carry-over).

45

23

45

23

5 3 = 15 Put down 5, “carry” 1.

45

23

5 1 is saved for later

45

23

5 1 is saved for later

4 3 = 125 2 = 10

12 + 10 + 1 = 23

Put down the 3 and “carry” the 2.

45

23

3 5

45 Have a “carry over” of 2.

23

35

4 2 = 8

8 + 2 = 10

Write in the 10.

45

23

103 5

Enough numbers for a while?

What do you see?

The brain has two hemispheres, each divided

into four lobes. Each lobe is responsible for

different functions.

• the frontal cortex is responsible for decision

making, planning, and some memory

• the temporal lobe (in green) for language

and memory

• the parietal lobe (in yellow) for spatial skills

• the occipital lobe is entirely devoted to

vision: It is thus the place where visual

illusions happen.

The visual system processes shapes, colors, sizes,

etc. One way to understand more about this

system is to look at how we can trick it, that is, to

look at how the brain reacts to visual illusions.

What do you see?

Optical IllusionsOur visual perception is created by our brain’s interpretation of

visual information entering through the visual pathway. The

visual cortex in your occipital lobe processes visual input from

your eyes. Not only is the occipital lobe mainly responsible for

visual reception, it also contains association areas that help in the

visual recognition of shapes and colors.

Inspiration for M C Escher?

Manipulative Puzzles and Pencil PuzzlesAgain, the occipital lobe contains association areas that help in

the visual recognition of shapes and colors. The problem solving,

selective attention, and executive functions used in these

exercises are handled by the anterior portion of the frontal lobes.

Example: Line up ten buttons in five rows with four buttons in

each row.

Miscellaneous Puzzles

Example: You’ve spent the last year building your home. Now

it’s time to add an important part to your project. When you go

to the hardware store to get what you need, the items are priced

as follows:

One will cost $5.

Two will cost $5.

Twelve will cost $10.

A hundred forty-four will cost $15.

What are these items?

1

2

12

144

From December 25, 2016 Parade Magazine:

The Power of the Puzzle

How do you keep the brain sharp? The key, says Marilyn vos

Savant, is to mix it up and try different ways to give your brain

a workout. “The most important part about staying sharp is

doing novel things,” she says. “It’s much better to solve 20

different problems every day than working at a computer doing

the same thing.” That may be one of the main reasons puzzles

remain so popular. “People like to exercise their brains,” she

says. “It gives them a feeling of accomplishment, that ‘Aha! I

did that – I got it right.’ It feels good and it’s good for you.”

Numbrix creator Marilyn vos Savant just so happens to have

recorded the highest IQ in the world -- an achievement that helped

lead to her gig at Parade Magazine. In the 1950s, at the tender age

of 10, the St. Louis-born vos Savant correctly answered every

question on the adult Stanford-Binet IQ test, which translated into

an astoundingly high IQ of 228. It wasn't until much later -- 1985,

in fact -- that she was recognized by Guinness World Records for

having the highest recorded IQ, a category Guinness eventually

removed because of worries that intelligence tests were too

disparate to allow for just one record holder.

59

65

71

73 77 37 31 1

3

5

9

1117475355

Complete 1 to 81 so the numbers follow a

horizontal or vertical path - no diagonals.

Numbrix:

Created by

Marilyn Vos Savant

30367874

22932353879767572

28333439806970

6232441426364

42726254081686766

8722214443626160

14152045505158

46 1013161949525756

1218485455 53 47 17 11

9

5

3

131377773

71

65

59

Grid Puzzles and Magic SquaresFactual memory is the ability to recall math facts; procedural

memory is the ability to recall steps/order needed to perform

mathematics. The left parietal lobe has been associated with

arithmetic fact retrieval and is the region involved with

processing numerical quantity.

Ken Ken: Each row and column contains the digits 1, 2, 3,

and 4. No repeats in any row or column. The arithmetic problem

tells you how the digits are related (what the answer is when using

the digits).

7+3+

2-

7+

4

2 3-

2

6X

Math Exercises for the Brain• Pattern Recognition

• Grid Puzzles

• Algebra and Computations

• Optical Illusions

• Logic Problems

Why exercise the brain?

Transience – memories degrade with the passing of time. This

can happen in sensory, short-term, and long-term storage. It

follows a general pattern where the information is rapidly

forgotten during the first couple of days or years, followed by

small losses in later days or years.

One of the key concerns of older adults is the experience

of memory loss, especially as it is one of the hallmark

symptoms of Alzheimer’s disease. However, memory loss is

qualitatively different in normal aging from the kind of

memory loss associated with a diagnosis of Alzheimer's.

Math Exercises for the Brain

• Manipulative Puzzles

• Pencil Puzzles

• Miscellaneous Puzzles

• Magic Squares

• Sam Loyd Puzzles

• Miscellaneous Brain Teasers

In which direction is the bus pictured below traveling?

The only possible answers are “left” or “right”.

In which direction is the bus pictured below traveling?

When pre-school children are shown this brain teaser, they often

answer “left.”

Why?

“Because you can’t see the door.”

Next, you’ll find some “quick questions”.

Go ahead, “blurt out” your answer!

Which month has 28 days?

What’s the easiest way to double your money?

What has a face and two hands but no arms or legs?

What 4 days of the week start with the letter ‘t’?

What goes up and doesn’t come back down?

If two’s company and three’s a crowd, what are four and five?

What has a thumb and four fingers but is not alive?

What occurs once in every minute, twice in every moment and yet

never in a thousand years?

What occurs twice in a week, once in a year but never in a day?

If there are 4 apples and you take away 3, how many do you

have?

Where do fish keep their money?

How did the soccer fan know before the game that the score

would be 0-0?

Which month has 28 days? All of them of course!

What’s the easiest way to double your money? Put it in front of a mirror.

What has a face and two hands but no arms or legs? A clock

What 4 days of the week start with the letter ‘t’? Tuesday, Thursday, today and tomorrow.

What goes up and doesn’t come back down? Your age.

If two’s company and three’s a crowd, what are four and five? 9

What has a thumb and four fingers but is not alive? A glove.

What occurs once in every minute, twice in every moment and yet never in a thousand years? The

letter m.

What occurs twice in a week, once in a year but never in a day? The letter e.

If there are 4 apples and you take away 3, how many do you have? You took 3 apples so obviously you

have 3.

Where do fish keep their money? In the river bank.

How did the soccer fan know before the game that the score would be 0-0? The score is always 0-0

before the game.

Three missionaries and three cannibals want to get to the other

side of a river. There is a small boat, which can fit only two. To

prevent a tragedy, there can never be more cannibals than

missionaries together.

How could all of them get safely across the river?

C

C

C

M

M

M

C

CC

M

M

M

First Trip Second Trip

C

C

CM

M

M

Third Trip Fourth Trip

C

C

C

M

M

M M

M

M C

C

C

Fifth Trip Sixth Trip

C

CC

MM

M M

M

M C

CC

Seventh Trip Eighth Trip

CC

C

MM

M

M

M

M

C

C

C

Ninth Trip Tenth Trip

C

C

C

M

M

M

M

M

M

C

C

C

Eleventh Trip

C

C

C

M

M

M

There are other correct

solutions. Can you find a

shorter solution?

If you had a 5-liter jug and a 3-liter jug, and an unlimited access

to water, how would you measure exactly 4 liters?

2

252

23

5

34

A Petri dish hosts a healthy colony of bacteria. Once a minute

every bacterium divides into two. The colony was founded by

a single cell at noon.

At exactly 12:43 (43 minutes later) the Petri dish was half

full.

At what time will the dish be full?

12:43 Half-full

12:44 Full

Math Exercises for the Brain

• Manipulative Puzzles

• Pencil Puzzles

• Miscellaneous Puzzles

• Magic Squares

• Sam Loyd Puzzles

• Miscellaneous Brain Teasers

Where do you find more puzzles?

• In the library• At the bookstore• Online