Puzzles Archive
This is a list of the previous puzzles that have been sent out by
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April 10, 2000
MindBender
Division
Find digits to replace A and B in "19A9B" so that the resulting five
digit number is divisible by 36. There is more than one solution.
___________________________
Mini-MindBender for Kids
Quads
Quadrilaterals are four sided polygons, with a square being one
example. Can you name four other figures that are quadrilaterals?
___________________________
...Answer to MindBender
Division
A number is divisible by 36 if it is divisible by 9 and 4. A number is
divisible by 4 if the lowest two digits are divisible by 4. Therefore,
B must be 2 or 6. In order for a number to be divisible by 9, the sum
of the digits must be divisible by 9. If B is 2, the sum of the digits
is 21+A. In this case A must be 6. If B is 6, the sum of the digits is
25+A. In this case A must be 2. Therefore the only solutions are 19692
and 19296.
This MindBender was modified from a puzzle in Robert M¸ller's book,
"The Great Book of Math Teasers."
___________________________
Answer to Mini-MindBender for Kids
Quads
Parallelogram, rectangle, rhombus, and trapezoid.
This MindBender was modified from a puzzle in Jerry Stickel's
"Mindbending Puzzles" calendar for 2000.
April 17, 2000
MindBender
Multiplication
Find a solution to the multiplication problem below.
I.e., replace A and B with any of the digits 1-9 (so there are no
leading zeroes in any number) and replace C and D with any of the
digits 0-9. A, B, C, and D are all different digits.
... AB
. X BA
______
. ACDA
___________________________
Mini-MindBender for Kids
River Crossing
Megan is returning from town with a fox, a duck, and a sack of corn.
She has to cross a river in a small rowboat that will only hold her
and one other thing. She decides that she must make more than one
trip. But she can't leave the fox alone with the duck because the fox
would eat the duck. She can't leave the duck alone with the corn
because the duck would eat the corn. How can Megan get everything
across the river safely?
___________________________
...Answer to MindBender
Multiplication
There are very few possibilities for the B and A combination that
result in a product ending in A.
B can be 1, with A being any other digit. The only values for A that
result in a four digit number are 7, 8, and 9, but none of these work
(71*17=1207; 81*18=1458; 91*19=1729). The first digit of the product
is not A.
B=3 and A=5, but 53*35=1855. The first digit of the product is not A.
B=5 and A=5, but 55*55=3025. The first digit of the product is not A.
B=6 and A=2, 4, 6, or 8. None of these work. The first digit of the
product is not A.
B=7 and A=5, but 57*75=4275. The first digit of the product is not A.
And finally B=9 and A=5. This works with 59*95=5605. So A=5, B=9,
C=6, and D=0.
This MindBender was modified from a puzzle in Jerry Stickel's
"Mindbending Puzzles" calendar for 2000.
___________________________
Answer to Mini-MindBender for Kids
River Crossing
Here's one solution:
1. Megan takes the duck across.
2. She returns by herself and then takes the fox across.
3. Megan returns with the duck, leaves the duck, and takes the corn
across.
4. She returns by herself and then takes the duck across.
This MindBender was modified from a puzzle in Raymond Blum's book,
"Math Trick, Puzzles & Games."
April 24, 2000
MindBender
Random Digits
In a large sample of random digits (0 to 9), what is the chance that
any random pair will sum to 15?
___________________________
Mini-MindBender for Kids
Allowance
Tom gets half the allowance that Mark gets. Suzie gets three times
what Tom gets. The three together receive $144. How much allowance
does each receive?
___________________________
...Answer to MindBender
Random Digits
The only two digit numbers whose two digits add to 15 are 69, 78, 87,
and 96. There are 100 possible two digit numbers (00 through 99).
Therefore, the chances are 4 out of 100 or 4%.
This MindBender was modified from a puzzle in Victor Serebriakoff's
book, "The Mammoth Book of Mindbending Puzzles."
___________________________
Answer to Mini-MindBender for Kids
Allowance
Tom gets X dollars in allowance. Mark gets twice as much or 2X dollars
in allowance. Suzie gets three times what Tom gets or 3X dollars.
Together they get 6X dollars. So, 6X=144 or X=24. Therefore, Tom gets
$24. Mark gets $48 and Suzie gets $72.
This MindBender was modified from a puzzle in Robert M¸ller's book,
"The Great Book of Math Teasers."
May 1, 2000
MindBender
What Number
What number when subtracted from its own square results in 42?
___________________________
Mini-MindBender for Kids
Change
You have a lot of change. You count it and discover three things: you
have a total of $12; you have only quarters (25 cents), dimes (10
cents), and nickels (5 cents); you have the same number of each. How
many of each do you have?
___________________________
...Answer to MindBender
What Number
The obvious answer is 7. However -6 is also a solution. The two
solutions both show up by factoring the following equation:
X*X-X-42=0
(X-7)(X+6)=0
X=7 or X=-6
This MindBender was modified from a puzzle in David J. Bodycombe's
book, "The Mammoth Puzzle Carnival."
___________________________
Answer to Mini-MindBender for Kids
Change
30 of each coin.
X*(25+10+5)=1200
X*40=1200
X=30
This MindBender was modified from a puzzle in Victor Serebriakoff's
book, "The Mammoth Book of Mindbending Puzzles."
May 8, 2000
MindBender
Two Leaders
You are choosing two leaders from a group of 625 people. You arrange
them in a 25 X 25 square. You tell the tallest person in each row to
step aside and then choose the shortest of those twenty-five as one
leader. Then all 25 return to their original places in the square. You
tell the shortest person in each column to step aside and then choose
the tallest of those twenty-five as the second leader. The two leaders
chosen by these two methods are different. Which one is taller?
___________________________
Mini-MindBender for Kids
Capital Letters
The following capital letters share a common characteristic
that is not shared with other capital letters in the alphabet:
B D E F H I K L M N P R T
Can you determine what that characteristic is? Hint: Think of how
these letters are constructed.
___________________________
...Answer to MindBender
Two Leaders
Let A be the first leader chosen and B be the second leader chosen.
Consider the following three cases: 1) If A and B were originally in
the same row, A would be taller. 2) If A and B were originally in the
same column, then B would be shorter, so A would be taller. 3) If they
were originally in different rows and columns, then consider the
person C who is in A's row and B's column. Then A is taller than C and
B is shorter than C (or C is taller than B). Therefore A is taller
than B.
So A is always taller than B.
This MindBender was modified from a puzzle in Pierre Berloquin's book,
"100 Games of Logic."
___________________________
Answer to Mini-MindBender for Kids
Capital Letters
These capital letters are the ones that have a full-height vertical
line.
This MindBender was modified from a puzzle in Jerry Stickel's
"Mindbending Puzzles" calendar for 2000.
May 15, 2000
MindBender
Missing Number
Can you logically complete the following square of numbers by
determining the missing number?
01 01 01 01
01 03 05 07
01 05 13 25
01 07 25 ??
___________________________
Mini-MindBender for Kids
Mowing
I hire a neighbor's son to mow my lawn. I want it cut every Saturday
for 15 consecutive weeks . The neighbor's boy agrees to do so. To make
sure he cuts my lawn every Saturday, I agree to pay him at the end of
the 15 weeks. The agreement states that I pay him $20 for every week
that he mows my lawn as long as he pays me $30 for every week that he
misses mowing my lawn. He agrees to this bargain and at the end of 15
weeks, we are even (he owes me as much as I owe him). How many weeks
did he miss?
___________________________
...Answer to MindBender
Missing Number
Each number is the sum of the numbers (if present) above itself,
left of itself, and above and left of itself. So the missing number
is 25+25+13 or 63.
This MindBender was modified from a puzzle in Pierre Berloquin's book,
"100 Games of Logic."
___________________________
Answer to Mini-MindBender for Kids
Mowing
Algebra helps us again. Let X be the number of weeks that he mowed.
20*X=30*(15-X)
20*X=450-30*X
50*X=450
X=9
So he cut the grass 9 weeks and missed 6 weeks.
This MindBender was modified from a puzzle in Adam Hart-Davis' book,
"Amazing Math Puzzles."
May 22, 2000
MindBender
7 Times
What is the smallest whole number, whose last digit (right-most) is
7, with the curious property that in order to multiply it by 7, all
that is needed is to take the 7 from its right end and move it to
the beginning of the number?
___________________________
Mini-MindBender for Kids
A Series Mistake
The following series has a mistake in it:
6 13 27 55 111 223 446 895 1791
Can you determine the rules and then spot the error?
___________________________
...Answer to MindBender
7 Times
1014492753623188405797.
Starting with 7*7=49, gives you 9 as the second digit in the original
number and a carry of 4 to the second product. Then 7*9+4=67 gives
you 7 as the third digit and a carry of 6 to the third product. Then
7*7+6=55 gives you 5 as the fourth digit and a carry of 5 to the
fourth product. Continue on in this fashion until the product is 7
with no carry to the next product. The resulting number is the
smallest number with this property. The next would be twice as long,
repeating all these digits to the left of the above answer or:
10144927536231884057971014492753623188405797.
This MindBender was modified from a puzzle on brainteasers network.
If you would like to subscribe please visit:
http://www.brainteasers.net/subscribe.html
___________________________
Answer to Mini-MindBender for Kids
A Series Mistake
The number 446 should be 447. Each number is twice the previous
number plus 1. Or the rule could be to add the next "power of two
multiple of 7" (1*7, 2*7, 4*7, 8*7, 16*7, etc.).
This MindBender was modified from a puzzle in Victor Serebriakoff's
book, "The Mammoth Book of Mindbending Puzzles."
May 29, 2000
MindBender
Division
Determine all the missing digits (shown as *s) in the following
division problem:
.............. **8**
......... ___________
..... *** | ********
............ ***
.......... _____
............. ****
.............. ***
............ _____
............... ****
............... ****
.............. _____
.................. 0
It looks more difficult than it really is.
___________________________
Mini-MindBender for Kids
5 Digit Number
A 5 digit number has the following properties:
1) The sum of the last two digits is one-half of the first digit.
2) The second digit is one-half of the first digit.
3) The fourth digit is one-half of the third digit.
4) The fifth digit is one-half of the fourth digit.
What is the number?
___________________________
...Answer to MindBender
Division
Since there are only three steps in the division, but five digits in
the quotient, the second and fourth digits in the quotient are 0. The
last digit in the quotient must be 9 since 8 would result in only a
three digit product as in the second step. Also, the divisor must be
smaller than 125 since 8 X 125 is a four digit number, and 8 X ***
is a three digit number. The first digit of the quotient must be
greater than 7 since even just 1000-(7*124)=132, too many digits for
the first subtraction. The first digit cannot be 9 since that results
in a four digit product, so the first digit is 8. Therefore, the
quotient is 80809. The divisor must be greater than 123 since
80809 X 123 is only seven digits. Therefore, the divisor is 124. The
whole division then becomes:
.............. 80809
......... ___________
..... 124 | 10020316
............ 992
.......... _____
............. 1003
.............. 992
............ _____
............... 1116
............... 1116
.............. _____
.................. 0
This MindBender was modified from a puzzle in Robert M¸ller's book,
"The Great Book of Math Teasers."
___________________________
Answer to Mini-MindBender for Kids
5 Digit Number
63421.
From 3) and 4), the third, fourth, and fifth digits must be 8 4 2, or
4 2 1. If they were 8 4 2, then from 1), the first digit would be 12
which is impossible. Therefore, the last three digits are 421. From 1),
the first digit is 6. From 2), the second digit is 3. So the number is
63421.
This MindBender was modified from a puzzle in Jerry Stickel's
"Mindbending Puzzles" calendar for 2000.
June 5, 2000
MindBender
Holes
Tim and two friends are digging identical holes in a field.
When Tim works with Ron, they dig 1 hole in 4 days.
When Tim works with Sam, they dig 1 hole in 3 days.
When Ron works with Sam, they dig 1 hole in 2 days.
When Tim works alone, how long does it take him to dig 1 hole?
___________________________
Mini-MindBender for Kids
Addition
Solve the following addition, determining which digits (0 to 9) are
replaced by the letters:
ABCDAEFF
AGHDFAFF
AAGDFAFF
__________
EHJKKJAA
___________________________
...Answer to MindBender
Holes
Let R, S, and T be the number of holes (or fraction of a hole) each
can dig in 1 day.
Then we have:
4T+4R=1
3T+3S=1
2R+2S=1
Solving, we get T=1/24. So Tim takes 24 days to dig 1 hole.
Another approach:
Let's imagine Tim's twin brother, Quin, who digs at Tim's rate. If
Tim and Ron, and Sam and Quin work together, in one day they dig
1/4 + 1/3 = 7/12 hole. Since Ron and Sam dig 1/2 or 6/12 hole in
one day, Tim and Quin dig 1/12 hole in one day. So Tim digs 1/2 of
1/12 or 1/24 hole in one day. So Tim can dig a hole in 24 days.
This MindBender was modified from a puzzle in Pierre Berloquin's book,
"100 Numerical Games."
___________________________
Answer to Mini-MindBender for Kids
Addition
Since A is 3*F with no carry and E is 3*A with no carry, F is 1,
A is 3, and E is 9. Then J is 5. And K is 6. So D is 2. This leaves
us with 4, 7, 8, and 0. B and G must be 4 0 or 0 4, since no carry
into the last column can occur. G must be 0 so C+H+G ends in 5 (15).
So G is 0, B is 4, resulting in C being 7 and H being 8, The total
solution is:
34723911
30821311
33021311
__________
98566533
I saw this addition problem recently but don't remember where.
June 12, 2000
MindBender
Word Family
The following words belong to a logical family:
AFGHAN
INDEFINITELY
SYNOPSIS
STUPENDOUS
BURST
Which of the following words belongs to the same logical family?
COMPANION
GLACIAL
HIJACKER
RESCRIPT
___________________________
Mini-MindBender for Kids
A Series
What is the next number in this series?
1 1 2 3 5 8 13 21
___________________________
...Answer to MindBender
Word Family
HIJACKER. Each word in the original group and "HIJACKER" have three
consecutive letters in alphabetical order.
This MindBender was modified from a puzzle in Pierre Berloquin's book,
"100 Games of Logic."
___________________________
Answer to Mini-MindBender for Kids
A Series
34. Each number (after the two starting 1s) is the sum of the
previous two numbers, so the next number in the series is 34=13+21.
This type of series is known as a Fibonacci series, after a famous
Italian mathematician who lived from about the year 1175 until 1230.
He introduced Algebra to Italy in the year 1202.
This MindBender was modified from a puzzle in Pierre Berloquin's book,
"100 Games of Logic."
June 19, 2000
MindBender
Four Glasses (A very difficult MindBender)
You are blindfolded and standing in front of a square table with a
drinking glass in each corner of the table. Each glass can be either
up or down, and as soon as all four are turned in the same direction
(either up or down) a bell will ring. On each turn you are allowed to
pick up any two of the four glasses and when you touch them you can
tell if they are up or down. You can then change the orientation of
either or both of these two glasses, if you wish. Then you complete
your turn by replacing the glasses in the same two corners of the
table where they were at the start of the turn. As soon as you set
both glasses down, the table will spin a random amount (and the bell
will ring if all four are turned the same way).
Determine a "minimum number of turns strategy" to guarantee all the
glasses are facing the same way (either all up or all down).
___________________________
Mini-MindBender for Kids
Martians
There are Martians that have 4 eyes, Martians that have 6 eyes,
Martians that have 8 eyes, and Martians that have 12 eyes. There is
an equal number of each type of Martian. The total number of eyes
that the Martians have between them is 5130. How many Martians of
each type are there?
___________________________
...Answer to MindBender
Four Glasses (A very difficult MindBender)
Number of turns required to guarantee the four glasses are all turned
in the same direction: Five.
The five step strategy:
1) Grab any two adjacent glasses. Turn them both up. If the bell
rings, you are done. If not, you are guaranteed to have two adjacent
ones up. The other two can be up or down.
2) Grab any two diagonal glasses. Turn them both up. If the bell
rings, you are done. If not, you are guaranteed to have 3 up and 1
down.
3) Grab any two diagonal glasses. They will either be both up, or
one up and one down. If you have the down one, turn it up. The bell
will HAVE to ring. If you have two up, turn one down. Now you are
guaranteed to have two up and two down in this pattern:
U D
U D or the same configuration rotated.
4) Grab any two adjacent glasses. If they are the same way, turn them
both the opposite way. The bell will HAVE to ring. If they are
opposite, reverse each of them. Now you are guaranteed to have the
following pattern:
U D
D U or the same configuration rotated.
5) Grab two diagonal glasses. They will be the same orientation (both
up or both down). Turn them the opposite way. The bell will HAVE to
ring and you are done.
This MindBender came from one of our MindBender solvers. I think it
is my favorite puzzle, even though I didn't find the beautiful
solution. Even if you didn't get this MindBender, it still is a very
beautiful and elegant puzzle and solution. And I think you can enjoy
it even if you didn't find the solution. Thanks very much to the
MindBender solver who shared this one with me!
___________________________
Answer to Mini-MindBender for Kids
Martians
171 Martians. 5130/(4+6+8+12) = 5130/30 = 171.
This MindBender was modified from a puzzle in Victor Serebriakoff's
book, "The Mammoth Book of Astounding Puzzles."
June 26, 2000
MindBender
Set Time
You don't own a watch, but you have an accurate clock that has
stopped because you have forgotten to wind it. You walk (at a
constant rate) to a friend's house (who also owns an accurate
clock) to spend the evening with him and walk home (at the
same constant rate) and accurately set your clock. How could
you do this without knowing ahead of time how long it takes to
walk to your friend's house?
___________________________
Mini-MindBender for Kids
Fast Multiply
Quick! What does 26 X 11 equal? 286. Now, how can you multiply any
two digit number by 11, very fast and without paper?
___________________________
...Answer to MindBender
Set Time
Wind and start your clock before you leave home and write down what
time it shows when you leave. Record the time it is when you arrive
at your friend's house (from his clock) and the time when you leave
his house to go home (again from his clock). When you get home, note
the time on your clock. From these four times, you know how long you
were away from home and how long you were at your friend's house.
Half of the diffence between these lengths of time is the time it
took you to get home. Adding this "time it took to get home" to the
time on his clock when you left your friend's house gives you the
current accurate time which you use to to set your clock.
This MindBender was modified from a puzzle in Raymond Smullyan's book,
"What Is the Name of This Book."
___________________________
Answer to Mini-MindBender for Kids
Fast Multiply
Add the two digits of the number together. Put that sum between
the two digits. So 26 X 11 = 286 since 2 + 6 = 8 and slipping that
8 between 2 and 6 gives you 286. If the sum of the two digits is
greater than 9, keep the units digit of the sum in the middle and
add 1 to the original left digit of the answer. For example,
76 X 11 = 836. 7 + 6 = 13, so slip 3 between 8 (7+1) and 6. This
may even result in a four digit answer. For example, 93 X 11 = 1023.
9 + 3 = 12, so slip 2 between 10 (9+1) and 3.
This MindBender was modified from a puzzle in Raymond Blum's book,
"Mathamusements."
July 3, 2000
MindBender
What Day
"L. T." Smith always lies on Mondays, Tuesdays, and Wednesdays.
"L. T." always tells the truth on all other days. "L. T." makes
the following two statements:
1. I lied yesterday.
2. I will lie again two days after tomorrow.
What day of the week is it?
___________________________
Mini-MindBender for Kids
Another Fast Multiply
Quick! What does 75 X 75 equal? 5625. Now, how can you multiply any
two digit number ending in 5 by itself, very fast and without paper?
Note: multiplying a number by itself is called "squaring" that number.
___________________________
...Answer to MindBender
What Day
If "L. T." is lying (i.e., it is Monday, Tuesday, or Wednesday),
statement 1 means it is Monday. If "L. T." is telling the truth (it
is Thursday, Friday, Saturday, or Sunday), statement 1 means it is
Thursday.
Statement 2 is possible if "L. T." is lying on Monday, but not
possible if "L. T." is telling the truth on Thursday.
Therefore, it is Monday.
This MindBender was modified from a puzzle in Raymond Smullyan's book,
"What Is the Name of This Book."
___________________________
Answer to Mini-MindBender for Kids
Another Fast Multiply
Multiply the first digit of the two digit number by one more than
itself. Then just put "25" after the answer. (The answer will always
end in 25.) In our example, 75 X 75, 7 X 8 = 56. Add 25 to the end
and you get 5625. 35 X 35 = 1225. 3 X 4 = 12 and put 25 after the 12.
This MindBender was modified from a puzzle in Raymond Blum's book,
"Mathamusements."
July 10, 2000
MindBender
Some Sum One
Replace letters with digits to make the following sum correct:
..... A
. + BBB
. + BBB
. + BBB
. + BBB
._______
.. ABBB
Try to find all solutions.
___________________________
Mini-MindBender for Kids
Memory
Amaze your friends by showing them you can "memorize" a very long
number. Write down 35831459437077415617853819 on a piece of paper.
Give this paper to your friend to hide from you. Then on another
piece of paper, rewrite the same number. You can even do this with
longer numbers. How do you do it?
___________________________
...Answer to MindBender
Some Sum One
It is obvious that the carry in each column must be A. Also, we
can see that 4B+A=10A+B or 3B=9A or B=3A. This only works if (A,B)
are (1,3) or (2,6) or (3,9). This gives us three solutions:
..... 1
. + 333
. + 333
. + 333
. + 333
._______
.. 1333
or
..... 2
. + 666
. + 666
. + 666
. + 666
._______
.. 2666
or
..... 3
. + 999
. + 999
. + 999
. + 999
._______
.. 3999
This MindBender was modified from a puzzle in Robert M¸ller's book,
"The Great Book of Math Teasers."
___________________________
Answer to Mini-MindBender for Kids
Memory
As you write the number down, just remember the first two digits and
how many digits long it is. Then after that, just add the two
previous digits to get the next digit, ignoring the tens digit of
the sum. Use this technique both as you write down the original
number and as you write down the number from "memory." Thus, in our
example, remember 35 and that the number is 26 digits long. Then
3+5=8, the third digit. 5+8=13, so the fourth digit is 3 (after
discarding the 1). Continue to get all 26 digits. If your friends
aren't amazed at first, repeat it with an even longer number starting
with two different digits.
This MindBender was modified from a puzzle in Raymond Blum's book,
"Mathemagic."
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