Math Exercises for the Brain
Sometimes it’s just as important to try as it is to actually get the right answer. It’s key to plan the steps toward a
solution and have a guess or estimate in mind.
Pattern recognition
(These puzzles work your executive functions in your frontal lobes by using your pattern recognition,
hypothesis testing, and logic.)
1) Odd man out: which is the odd man out in each row? (There may be more than one answer.)
36
42
45
72
18
10
21
18
61
25
35
63
49
16
56
2
3
5
10
12
2) Find the missing number in the last triangle.
3) When you divide 12 by 5, the remainder is 2; it’s what’s left over after you have removed all the 5s from the
12. When you raise 4 to the fifth power (that is, 45), you multiply four by itself five times: 4x4x4x4x4, which
equals 1,024. What is the remainder when you divide 100100 by 11?
4) What comes next in each sequence?
a) 4, 7, 12, 19, 28, ___
b) 6, 20, 27, 41, 48, ___
c) 23, 16, 10, 5, 1, ___
d) 9, 18, 54, 216, 1080, __
e) 1, 1, 2, 3, 5, 8, 13, 21, 34, ___
f) 2, 5, 10, 17, 26, 37, ___
g) o, t, t, f, f, s, s, ___
5) Determine the relationship between the numbers in each of the first three sets of numbers, and then apply that
to the last three sets to fill in the blanks.
2
5
10
3
10
20
4
17
34
5
__
__
__
37
__
7
__
100
6) What is the logic behind this sequence, and why can’t it be continued?
0, 1, 8, 10, 19, 90
7) What do the following numbers have in common?
3
7
10
11
12
17
8) What number should replace the question mark in the following?
5
9
17
13
25
49
37
73
?
Grid Puzzles
(Working memory: the ability to keep information in your mind while working on integrating
and processing it. This involves the frontal lobes.)
1) Ken Ken
2
12X
2-
2-
3+
3X
3-
7+
2
3+
12X
24X
3
2-
2
4+
3
3-
20X
15X
3-
9+
3+
3
11+
24
9+
2) Sudoku
96X
25X
2
3+
2-
3X
14+
18X
1-
2
9+
2
4+
2
1-
4
6+
5-
36X
3-
7+
3
8+
3) Numbrix
Fill the grid with consecutive numbers in a horizontal or vertical sequential path.
7
9
6
16
13
3
25
23
1
19
35
36
22
33
4) Domino Hunt
5) Complete the figure logically.
30
6) Insert the missing number.
10
7
8
11
5
3
6
7) What goes in the empty square?
Algebra and Computations
(Factual memory is the ability to recall math facts; procedural memory is the ability to recall steps/order
needed to perform mathematics.)
1) Four numbers
Use four 4’s in computations that result in 1, 2, 3, …, 10.
1
44
44
2
4 4
44
3
444
4
4
4 4
44
5 4 4 
4
4
etc.
Then use four 5’s or four 3’s – any digits except 0 or 1.
2) Fill in the 9 digits: 1, 2, 3, 4, 5, 6, 7, 8, 9 into the boxes to make the equation correct. Put all the odd numbers
on the left and even numbers on the right.
3) Use the numbers 1 - 9 to fill in the bubbles so that the sum around each hexagon adds up to 30. You can use
the numbers more than once, and you won’t use all of them. (There is more than one answer.)
4
1
6
2
4
9
4
7
6
4) Each letter represents a distinct (different) numeric digit. If the sum is correct, what digit does each letter
represent?
FOOD
+
FAD
DI ET S
5) Each letter represents a distinct (different) numeric digit. If the sum is correct, what digit does each letter
represent?
LPNLP
×
L
AAAAAA
6) Using all the digits, 1, 2, 3, 4, 5, 6, 7, 8, 9, above and below a fraction symbol, create a fraction equaling 1/3.
7) If TRAIN = 12954, find the four numbers that represent CHOO when
CHOO
+CHOO
TRAIN
8) There’s a two-digit number that, when read from right to left, is 4
1
times as large as from left to right. What
2
is the number?
9) Not counting two dollar bills, what’s the most money you can have without being able to change a $20 bill?
10) A ruler has only four marks on it, but it can still measure the length of any whole number from 1 to 12
inches. One of the marks is 1 inch from the end of the ruler. Where are the others?
11) You’ve got a 7-minute hourglass and an 11-minute hour-glass. What’s the quickest way to time the boiling
of some pasta for 15 minutes?
12) Replace each letter for a unique digit so that the equation is correct.
(J+O+I+N+T)3 = JOINT
Optical Illusions
(Our visual perception is created by our brain’s interpretation in the cerebral cortex of visual information
entering through the visual pathway. Our minds get actively involved in interpreting the perceptual
input, rather than passively, objectively recording it…please remember that next time you have a strong
disagreement with someone. You may both be facing the same exact reality but “seeing” it in different
ways!)
1) Are the horizontal lines straight or crooked?
2) How would you describe the inner shape?
3) Describe the figure in two words.
4) How many cubes?
5) How many legs?
6) How many boards?
7) Which middle segment is longer?
8) Which man is smallest?
Logic Problems
1) There are 3 gentlemen in a meeting: Mr. Yellow, Mr. Green and Mr. Brown. They are wearing yellow, green
and brown ties. Mr. Yellow says: “Did you notice that the color of our ties are different from our names?” The
person who is wearing the green tie says, “Yes, you are right.” Who is wearing what color of tie?
2) The police department arrested four suspects – two men and two women – on suspicion of petty theft. The
sergeant on duty who processed the suspects was having a bit of a bad day. He produced this list of suspects and
descriptions:
Robin Wilde: scar on left cheek
Cary Steele: purple hair
Pat Fleece: tall and blonde
Connie Theeves: birthmark on left wrist
When the list landed on the arresting detective’s desk, he was furious. He went to the sergeant and said, “Paul,
you might be having a bad day, but this is full of mistakes. The first and last names are all mismatched. And
none of the descriptions matches either the first or the last name it is listed with.
The sergeant replied, “Sorry, I am having a bad day. But I think I need a little more information to fix this.”
The detective answered, “Okay. Here’s more info.”
Connie has purple hair to match her purple high tops.
The men are Steele and Fleece.
A woman has the scar.
The sergeant then determined the first and last names of each suspect – as well as the descriptions.
3) Two men (Jack and Mike) and two women (Adele and Edna) each like a different type of music (one likes
jazz). Their last names are Mullin, Hardaway, Richmond, and Higgins. Find each person’s full name and
favorite type of music.
Hardaway hates country-western music.
The classical-music lover said she’d teach Higgins to play the piano.
Adele and Richmond knew the country-western fan in high school.
Jack and the man who likes rock music work in the same building.
Richmond and Higgins are on the same bowling team. There are no men on their team.
4) Three friends – Elaine, Kelly, and Shannon – all start for their college volleyball team. Each plays a different
position: setter, middle blocker, and outside hitter. Of the three, one is a freshman, one a sophomore, and the
other a junior. Determine each woman’s position and year in school.
Elaine is not the setter.
Kelly has been in school longer than the middle blocker.
The middle blocker has been in school longer than the outside hitter.
Either Kelly is the setter or Elaine is the middle blocker.
5) A lady and a gentleman are sister and brother. We do not know who is older.
Someone asked them: “Who is older?”
The sister said: “I am older.”
The brother said: “I am younger.”
At least one of them is lying. Who is older?
6) The aisles in Mary’s Mini-Mart are numbered one to six, starting at the entrance. Cleaning products are in the
aisle next to beverages, and aren’t the first items you see when entering. The meat aisle is closer to the entrance
than the bread aisle. Cereals are two aisles before beverages, and meat is four aisles after produce. How are the
aisles arranged?
Manipulative Puzzles
(The parietal lobes are involved in visual interpretations.)
1) Arrange the thirteen toothpicks as shown. This arrangement could be used to house six rabbits in six little
hutches. Now take away one toothpick and use only twelve to create a new arrangement of six little hutches for
the rabbits.
2) Move 1 toothpick to correct the equation.
3) Starting with the arrangement of ten toothpicks as shown, create each of the following:
a) Move four matches so that three squares are created.
b) Move three matches so that two rectangles are created.
c) Move two matches so that two rectangles are created.
4) Move just 3 toothpicks so the fish swims in the other direction.
5) Move one stick to make a square.
Pencil Puzzles
1) Draw the picture without picking up your pencil or tracing over any lines.
2) Plant 7 rosebushes so that they form 6 straight lines with 3 rosebushes in each line. Or plant 10 bushes in 5
lines with 4 in each.
3) Connect all nine points with exactly 4 connected straight lines without lifting your pencil off the paper.
Miscellaneous Puzzles
1) Bridge Crossing
A group of four people has to cross a bridge. It is dark, and they have to light the path with a flashlight. No
more than two people can cross the bridge simultaneously, and the group has only one flashlight. It takes
different time for the people in the group to cross the bridge:
Annie crosses the bridge in 1 minute,
Bob crosses the bridge in 2 minutes,
Carlie crosses the bridge in 5 minutes,
Dorothy crosses the bridge in 10 minutes.
How can the group cross the bridge in 17 minutes?
2) A number of children are standing in a circle. They are evenly spaced, and the 7th child is directly opposite
the 17th child. How many children are there, altogether?
3) In a family photo, the following are in the picture:
1 grandfather, 1 grandmother, 2 fathers, 2 mothers, 6 children, 4 grandchildren, 2 sisters, 2 brothers, 3 sons, 3
daughters, 1 father-in-law, 1 mother-in-law, 1 daughter-in-law.
What is the fewest number of people possible in the picture?
4) You have a boat and need to take a fox, a chicken, and some corn across the river. The boat will only hold
you and one other thing. If you leave the chicken alone with the corn, the chicken will eat the corn. If you leave
the fox alone with the chicken, the fox will eat the chicken. How can you get them all across the river with the
least amount of trips necessary?
5) What’s the rule that governs the change in the position of both the gray squares and the black squares from
Figure A to Figure B? When you have the answer, use the rule to fill in Figure C.
A
B
C
6) The diagram below shows a pattern made up of squares. How many squares can be found?
Magic Squares
(Each horizontal, vertical, and diagonal adds up to the same amount.)
8
3
4
1
5
9
6
7
2
1) Complete a magic square using the numbers 3, 6, 9, 12, 15, 18, 21, 24, 27.
2) Complete a magic square using the numbers 2, 4, 6, 8, 10, 12, 14, 16, 18
Sam Loyd Puzzles
(Samuel Loyd (January 31, 1841 – April 10, 1911) was perhaps America’s greatest ever puzzle maker,
inventing and refining thousands of puzzles in his lifetime, including word puzzles, rebuses, tangrams
and math puzzles.)
1) The Covent Garden Puzzle
(This puzzle appeared in London half-a-century ago, accompanied by the somewhat surprising assertion that it
had mystified the best mathematicians of England.)
Mrs. Smith and Mrs. Jones had an equal number of apples, but Mrs. Jones had larger fruits and was selling hers
at the rate of two for a penny. Mrs. Smith, whose apples were smaller, sold hers at the rate of three for a penny.
Mrs. Smith was for some reason called away and asked Mrs. Jones to dispose of her stock. Upon accepting the
responsibility of disposing of her friend’s stock, Mrs. Jones mixed the apples together and sold them all at the
rate of five apples for two pence. When Mrs. Smith returned the next day, the apples had all been disposed of,
but, when they came to divide the proceeds, they found that they were seven pence short. Supposing that they
divided the money equally, each taking one-half. How many apples were sold, and who was “shorted” the seven
pence?
2) The Columbus Problem Puzzle
The dot over a number signifies that it is a repeater, which would go on forever. For example, to describe the
.
. .
1
3
fraction , which is 0.33333…, you would use .3 . To describe
, which is 0.272727…, you would use .2 7 .
3
11
One very nice property is that any proper fraction with just 9’s in the denominator has the numerator repeating
23
456
exactly as written:
 .232323...,
 .456456456...
99
999
The solution to this problem uses eight “dots” and the digits 4, 5, 6, 7, 8, 9, 0.
3) Sam Loyd’s Archery Puzzle
At a recent archery meeting, the young lady who won the first prize scored exactly one hundred points. The
scores on the target were: 16, 17, 23, 24, 39, and 40. How many arrows did she use to get her score?
Miscellaneous Brain Teasers
1) I am an odd number. Take away one letter and I become even. What number am I?
2) Using only addition, how do you add eight 8’s and get the number 1000?
3) Sally is 54 years old and her mother is 80, how many years ago was Sally’s mother three times her age?
4) Which 3 numbers have the same answer whether they’re added or multiplied together?
5) There is a three digit number. The second digit is four times as big as the third digit, while the first digit is
three less than the second digit. What is the number?
6) The day before yesterday I was 25 and the next year I will be 28. This is true only one day in a year.
What day is my birthday?
7) Eight children divided 32 apples as follows: Angela got one, Marcy two, Janette three, and Kristen four.
Norbert Smythe took as many as his sister. Tristan Brownfield took twice as many as his sister. Bart Jackson
took three times as many as his sister, and Jed Redfield took four times as many as his sister. What are the girls’
last names?
8) If Sybil has the same number of sisters and brothers, but each of her brothers has only half as many brothers
as sisters, how many sisters and brothers are there in the family?