180 Days of
Math Tasks
Low-Floor, High-Ceiling Engagement
Curated by Dan Shuster
@DanShuster on TwiX and BlueSky
All sources noted on each slide
1. 16 Objects, 4 Piles
How can 16 objects be put into four piles so that each pile has a different number of objects in it? How many ways are possible?
Here is one possibility: (1, 2, 3, 10)
Source: Adapted from Marilyn Burns, marilynburnsmathblog.com
2. How Much Money?
Your friend tells you that they have two bills in their pocket and the bills can only be $1, $5, $10 or $20 bills. What are the possible amounts, in dollars, that these two bills could consist of? Note: both bills can be the same.
Source: Dan Shuster, @DanShuster on TwiX
Extension: what if the bills can’t be the same?
3. Seed Numbers
Consider two whole numbers (for example 2 & 7). These will be the first two numbers. The third number is the sum of the first two (9). The fourth is the sum of the previous two (16), and so on (2, 7, 9, 16, 25, 41, …). What do the first two number have to be so that the fifth number is 100?
Source: Adapted from Peter Liljedahl, peterliljedahl.com/teachers/good-problem
4. Divide and Conquer
Take the digits 1,2,3,4,5,6 and make two 3-digit numbers from them. Multiply them together. What do you get? Try to find the pair of 3-digit numbers that will make the largest product.
Example: 132 x 546 = 72,072
Source: Adapted from Fawn Nguyen, fawnnguyen.com/teach/multiplication-finding-the-greatest-product
5. How Many Squares?
How many squares do you see in the diagram at right?
6. Number Pyramids
Start with the numbers 1-5 at the bottom in any order - then each of the squares above will be the sum of the two numbers below it (see an example at right). What arrangement of the numbers 1-5 at the bottom creates the highest number in the top square of the pyramid?
Source: Adapted from transum.org/Software/SW/Starter_of_the_day/students/pyramid
2
3
1
5
4
4
3
7
9
7
10
16
17
26
43
7. Six or Eight?
If you roll two standard 6-sided dice and add the numbers together, are you more likely to get a sum of 6 or a sum of 8?
8. Reach 100
Source: nrich.maths.org/1130
Reach 100
Your challenge is to find four different digits that give four two-digit numbers which add to a total of 100.
See an example at right.
9. Chicken Nugget Math
Suppose that Chicken Nuggets only come in packs of size 4 and size 9. Determine which numbers of Nuggets from 1 to 25 are possible to buy from these size packs. For example, you CAN buy 17 (2 4-packs and 1 9-pack). You CANNOT buy 6 nuggets though.
10. Chicken Nuggets, Revisited
Suppose that Chicken Nuggets only come in packs of size 6, 9 and 20. Determine why 43 nuggets is the largest number that CANNOT be bought when combining these various boxes.
11. How Many Cannonballs?
Source: Adapted from brainly.com/question/5056521
How many cannonballs are in the picture at right? How can you use your findings to generalize for any such pyramid stack of cannonballs?
12. How Many 7’s?
If you write out the numbers from 1 to 1000, how many times will you write the number 7?
13. Change for a Dollar
Source: Adapted from maa.org/frank-morgans-math-chat-293-ways-to-make-change-for-a-dollar
How many ways can you make a dollar using any combination of nickels, dimes and/or quarters? Write down as many solutions as you can in the time provided.
14. The 4th of July
Source: Dan Shuster, @DanShuster on TwiX
Suppose that in a given year, January 1st is on a Tuesday. What day of the week would July 4th be on in that year? Assume that the year is not a Leap Year, so there is not a February 29 to consider. Recall that February has 28 days in regular years, and 29 days in Leap Years.
15. Magic V
Source: nrich.maths.org/6274
Place each of the numbers 1 to 5 in the V shape at right so that the two arms of the V have the same total. Find multiple solutions!
16. Smallest and Largest
Source: fawnnguyen.com/teach/smallest-largest Handout:: Google Doc
Using the handout provided, complete the given task.
17. Three Consecutive Integers
Source: unknown - see this Google Doc for explanation of investigation
Then try numbers spaced 2 apart, 3 apart…
18. Four Consecutive Integers
Four Consecutive Numbers: a, b, c, d
Take any four consecutive numbers - for example 3, 4, 5, 6.
19. Close to 10,000 - Open Middle
Using the digits 1-9 only once each, create two factors that will result in a product as close to 10,000 without going over.
20. Ice Cream Scoops
In shops with lots of ice cream flavors, there are many different flavor combinations, even with only a 2-scoop cone. With 1 ice-cream flavor there is 1 kind of 2-scoop ice cream, but with 2 flavors there are 3 possible combinations (eg vanilla/vanilla, chocolate/chocolate, and vanilla/chocolate). How many combinations are possible with 3 flavors? 4 flavors? 5 flavors? Use your results to find a pattern to help you find the answer for 10 flavors.
21. Sum of 51
How many 6 digit numbers are there whose digits sum to 51? Write down as many as you can in the time allowed. And… go!
22. What’s the Perimeter?
Source: Sunil Singh, @mathgarden on TwiX
23. How Many Ways?
Source: Adapted from gdaymath.com/lessons/perms-1/1-2-word-games
24. Square Sum Challenge
Source: Adapted from youtube.com/watch?v=G1m7goLCJDY&t=2s
25. Chickens and Pigs
Source: Adapted from youtube.com/watch?v=RJKk0Q6H5dI
A farmer has some chickens and some pigs. The farmer notices that their animals have a total of 22 legs. How many chickens and how many pigs might they have?
26. What’s Next?
Source: Adapted from various sources
Determine a pattern which will help you find the next two numbers in each sequence:
27. The Four 4’s
Source: youcubed.org/tasks/the-four-4s
Can you make every number between 1 and 20 using exactly four 4’s and any math operations?
Here are some examples:
4 + 4 + 4 ⁄ 4 = 9
4 x 4 - 4 + 4 = 16
(44 - 4!) / 4 = 5
28. Patterns and Products
29. Leo the Rabbit
Source: youcubed.org/tasks/patterns-and-products
Handout: youcubed2.wpenginepowered.com/wp-content/uploads/2017/03/Leo-the-Rabbit-handout2.pdf
Leo the Rabbit is climbing up a flight of 10 steps. Leo can only hop up 1 or 2 steps each time he hops. He never hops down, only up. How many different ways can Leo hop up the flight of 10 steps? Provide evidence to justify your thinking.
30. Largest Product
Here are some different ways in which we can split 100:
The products of these sets are all different:
What is the largest product that can be made from whole numbers that add up to 100?
31. Elevenses
In the grid below at right, look for pairs of numbers that add up to a multiple of 11.
Are there any numbers that can only have one partner?
Are there any numbers that could have more than one partner?
What is special about numbers which have the same set of partners?
Can you find every possible pair?
How can you be sure you haven't
missed any?
Source: nrich.maths.org/6390
32. NOT Perfect Squares
Without using a calculator, how can you quickly determine which of the following numbers are definitely NOT perfect squares?
123456 378450
457233 514767
712442 974551
658149 231668
Source: Adapted from burningmath.blogspot.com/2013/09/how-to-check-if-number-is-perfect-square.html
33. The Ones Digit
Without using a calculator, can you determine what the ones digit of this calculation is?
1100 + 2100 + 3100
Source: Adapted from corbettmaths.com/wp-content/uploads/2014/08/february-20.pdf
34. A and B
Given that A and B are positive integers where A<B, solve:
A + AB + B = 328
Source: Chris Smith’s Mathematics Newsletter, Issue #646, August 21, 2023 | @aap03102 on TwiX
35. Connect the Dots
How many triangles can be drawing using points from the grid at right as vertices?
36. 1,001 Pennies
There are 1001 pennies lined up on a table. I come along and replace every 2nd coin with a nickel. Then, replace every 3rd coin with a dime. Finally, I replace every 4th coin with a quarter. How much money is on the table?
Source: brainly.com
37. Twice the Product
Can you find a two-digit number that is equal to two times the product of its digits? As a non-example: if we try 46, 4 x 6 = 24 - now multiply that by 2 and we get 48. So close, but not 46!
Source: Adapted from corbettmaths.com/wp-content/uploads/2014/08/march-8.pdf
38. Thrice the Product
Can you find two two-digit numbers that are equal to three times the product of their digits? As a non-example: if we try 32, 3 x 2 = 6 - now multiply that by 3 and we get 18. Sorry, not 32!
Source: Adapted from corbettmaths.com/wp-content/uploads/2014/08/april-8.pdf
39. Pair and Share
The words “zero” and “one” share letters (e and o). The words “one” and “two” share a letter (o), and the words “two” and “three” also share a letter (t). How far do you have to count in English to find two consecutive numbers that don’t share a letter in common?
Source: Chris Smith and Alex Bellos - Summer 2023 Puzzle Supplement - theguardian.com
40. Pair of Integers
41. How Many Triangles?
How many triangles are shown in the diagram below?
42. Intersecting Lines
What is the maximum number of points of intersection that can be determined by ten straight lines?
3 intersecting lines
43. Powers of 5
What are the last 3 digits of 51000?
51000
44. Cryptarithms 1
In the puzzles below, each letter stands for a different digit (0 is never the first digit of any number). Find each solution. Do any of them have more than one solution?
Source: nrich.maths.org/cryptarithms
45. Reverse the Digits
Compute 26 x 93 and 62 x 39
Find two more pairs of multiplications with the same property.
If ab x cd = ba x dc, state a relationship between a, b, c and d.
46. How Many Numbers?
How many three digit numbers have all digits different?
How many four digit numbers have all digits different?
How many five digit numbers have all digits different?
Can you generalize this for any number of digits?
47. Put Them in Order
Arrange in order from lowest to highest:
288 | 355 | 544 | 733 |
48. What’s the Mean?
Four numbers are written in a row.
The mean of the first two is 5.5
The mean of the middle two is 6.5
The mean of the last two is 2.5
Find the mean of the first and last numbers.
Source: Adapted from corbettmaths.com/wp-content/uploads/2014/08/september-10.pdf
49. What’s the Digit?
Find the units digit of
2121 + 2424 + 2626
50. How many only take Algebra?
In a collection of 170 students, 108 of them take Spanish, 91 take Algebra and 11 take neither of these courses. How many of these students ONLY take Algebra (so not Spanish).
51. Sequence
In the sequence
a, b, c, d, e, f, g, h, i, j, k, 0, 1, 1, 2, 3, 5, 8, 13…
each term is the sum of the two terms to its left.
Find the value of a.
52. Find x
53. Cryptarithms 2
Source: nrich.maths.org/cryptarithms
In the puzzles below, each letter stands for a different digit (0 is never the first digit of any number). Find each solution. Do any of them have more than one solution?
54. Triangles in a Square
The figure at right consists of a square divided into 5 triangles. Find all missing lengths.
Source: Chris Smith’s Mathematics Newsletter, Issue #587, @aap03102 on TwiX
55. One Million Seconds
How long is one million seconds? Break it down to days, hours, minutes and seconds.
56. Soldiers and Loaves
57. What’s the Price?
Find the missing price
58. Double Up!
How many fractions can you find whose value is doubled when 3 is added to both the numerator and the denominator of the fraction.
59. Unit Fraction Sum
Find pairs of integer values m and n such that:
60. Triangles in a Grid
How many triangles can be drawn having points in the grid at right as vertices?
61. A Group Gift
A group of people are buying their friend a birthday gift which has a fixed cost.
If each person puts in $9, they have $11 leftover.
If each person puts in $6, they will be $16 short.
Find how much the gift costs and
how many people are in the group.
62. Triple Double
What do you notice?
Why does this happen?
63. Mystery Multiplication
What do you notice?
Why does this happen?
Devise a similar process using what you have learned.
64. Prime Triangles
Fill in each circle with a prime number so that the sum of the large triangle is 20 and each small triangle has the same sum.
65. Stick-y Triangles
In a sack, there are seven sticks with lengths
2cm, 4cm, 5cm, 8cm, 10cm, 11cm and 19cm.
If three sticks are picked at random, what is the probability that they can form a triangle?
66. Squares for the Ages
If you add the square of Chelsea’s age to the age of Jamie, the sum is 81. If you add the square of Jamie’s age to the age of Chelsea, the result is 297. Find their ages.
Then, make your own version of this question.
67. Cryptarithms 3
Source: nrich.maths.org/cryptarithms
In the puzzles below, each letter stands for a different digit (0 is never the first digit of any number). Find each solution. Do any of them have more than one solution?
68. Multiplying Large Numbers
Source: Based on a problem from openmiddle.com
Using the digits 1 to 9, at most one time each, create two 3-digit numbers that have a product as close to 500,000 as possible.
69. Prime Numbers
Source: openmiddle.com/prime-numbers
Use the digits 1 to 9, at most one time each, to make 5 prime numbers. How many different solutions can you find?
70. Goldbach Conjecture
The Goldbach Conjecture is a yet unproven conjecture stating that every even integer greater than two is the sum of two prime numbers. For example, 12 = 5 + 7. Try to find two different ways to express each of the following numbers as a sum of two primes.
18 32 42 52 64 78 84 96 112 128
71. Magic Triangles
Put the digits 1-6 in the circles below, using each number once, so that each side of the triangle adds up to the number given.
72. Geometry Puzzler
Source: unknown
For the shape given, the area and perimeter are numerically equal. Find the value of that will make this true.
73. Vedic Square
Source: wikipedia.org/wiki/Vedic_square - inspired by Sunil Singh (@Mathgarden) in Math Recess
In Indian mathematics, a Vedic Square is a variation on a typical multiplication table. Can you figure out how the numbers in the table/square are determined?
74. 1 to 9 Puzzle
Source: @1to9Puzzle on TwiX/X
For each puzzle, fill the blank squares with 8 digits from 1-9, using each digit only once. The central square is not used. The numbers in the jutting squares represent the products of the three digits in the corresponding row/column.
75. Equivalent Fractions
Use the digits 1 to 9, at most one time each, to make three equivalent fractions. There are three possible answers - how many can you find?
76. Bovine Math 1
Source: Peter Harrison
In the diagrams at right, the number on each bridge is the sum of the numbers of cows (circles) in each of the adjoining fields. In the 3 of the problems, there is an additional hint. Find all of the unknown numbers.
77. Bovine Math 2
Source: Peter Harrison
In the diagrams at left, the number on each bridge is the sum of the numbers of cows in each of the adjoining fields. One of these two puzzles is solvable and the other is not. Which and why? Find all of the unknown values in the solvable puzzle.
78. Bovine Math 3
In the diagrams at left, the number on each bridge is the difference of the numbers of cows (circles) in each of the adjoining fields. In the 3 of the problems, there is an additional hint. Find all of the unknown numbers.
79. Area Mazes
Prolific Japanese puzzle inventor Naoki Inaba created these puzzles. The goal is to find the missing value using the simple concept of area of a rectangle - which is the length multiplied by the width.
80. Matchstick Math
How can you make the equation true by moving ONLY ONE matchstick? There are at least 3 ways to do this! How many can you find?
81. Wacky Multiplication Table
Source: dobmathsenrichment.weebly.com
Complete this wacky multiplication table and find the values of a, b, c, d, e, f, g and h. Then create your own version of this puzzle and share.
82. Equation Puzzle
Source: Erich’s Puzzle Palace - erich-friedman.github.io/puzzle/number
Each of the numbers from 1-8 must be placed in the white squares below so that all of the vertical and horizontal equations are true. Use each number exactly once.
83. Five Cards
84. 1 to 10 Puzzle
Source: @1to9puzzle on TwiX
In this puzzle, fill in the empty yellow squares with the numbers from 1 to 10, using each number exactly once. The numbers in the green squares represent the sums of the digits in their yellow adjacent squares. Adjacent squares share an edge with another square.
85. Fraction Action
Source: fractiontalks.com, Nat Banting (@NatBanting on TwiX)
What fraction of each shape is shaded?
What fraction of the square is blue? Yellow? Red?
What fraction of the square is blue?
Yellow? Green? Red?
What fraction of the square is blue?
86. Yohaku Puzzle
Source: Yohaku.ca, @YohakuPuzzle on TwiX
Fill in the cells in the table with 9 different integers to get the products shown in each row and column. For example, the three numbers in the middle row multiply to be -6.
87. Open Middle Square Root
Using the digits 1 to 9, at most one time each, fill in the boxes to make the following expression as close to 0 as possible.
88. 0 to 9 Frame
Source: Jordan Rappaport, @JRappaport27 on TwiX
Use the digits 0 to 9, once each, to fill the ten squares of the “frame” so that the top row, bottom row, left column and right column all add up the same sum. There are 480 ways to do this! How many can you find?
89. Arithmetic Sequences
Source: mathshko.com/2018/09/24/linear-sequences
Complete the two puzzles below:
90. Algebraic Puzzlers
Source: YouTube and mindyourdecisions.com
Complete the two puzzles below:
91. Would You Rather?
Source: wouldyourathermath.com
Use math (and some research) to justify your answer to this question:
92. Conjecture and Proof
Source: @Matematickcom on TwiX
A conjecture is a prediction based on evidence. A proof shows why that prediction is correct. Make a conjecture about the following. Test out your conjecture and see if you can prove why it always works.
(1 x 9) + (1 + 9) = 19
(2 x 9) + (2 + 9) = 29
(3 x 9) + (3 + 9) = 39
etc...?
93. Area and Perimeter
Source: Dan Shuster, @DanShuster on TwiX, BlueSky
Find two different rectangles with single-digit length and width to satisfy each of the following situations:
What conjectures can you make about the length and width of the rectangles that satisfy each of these conditions?
94. Coins vs Dice
Source: wouldyourathermath.com
Consider the following two games of chance:
Game 1 - You win $100 if you flip three coins and have them all end up the same (all heads or all tails).
Game 2 - You win $100 if you roll three 6-sided dice and have them all end up on different numbers.
If you get only one try at either game to win, Would You Rather play Game 1 or Game 2?
95. Open Middle 2-Step Equation
Source: Adapted from openmiddle.com/two-step-equations
Using the digits 1 to 9 at most one time each, place a digit in each box to create an equation in which the solution for x is 2 (so x = 2 is the answer to the equation). There are 13 different ways to do this - how many can you find?
96. Square Root = Fraction
Source: Dan Shuster, @DanShuster on TwiX and BlueSky
There are 3 pairs of numbers a and b less than 100 that satisfy the following equation:
See how many of these 3 pairs you can find. Can you determine a relationship between a and b that would allow you to find infinitely more solutions?
97. Pyramid Puzzle
Fill in the spots at right with the numbers 0 - 10, so that each number in the pyramid is the sum of the two below it. You may use numbers more than once. How many different ways can you solve this puzzle?
98. 1 to 10 Product Pyramid
Source: @1to9Puzzle on TwiX
Fill in the empty yellow squares with the numbers from 1 to 10, using each number exactly once. The numbers in the green squares represent the products of the digits in their yellow adjacent squares.
99. Sum of Two Squares
Can all integers be written as the sum of two squares? In number theory, the sum of two squares theorem says that for certain integers, n > 1 that n can be written as a sum of two squares. That is, n = a2 + b2 for some integers a, b.
Determine which integers from 1 to 20 can be written as a2 + b2. The list has been started for you below:
1 = 02 + 12 2 = 12 + 12
100. Funny Factorizations
Some 4-digit numbers can be written as the product of a 3-digit number and a 2-digit number using all of the digits 1 to 9 each once and only once among them.
For example, 4396 = 127 x 58 (all nine digits used once each)
The numbers 5346 and 5796 can each be expressed as a product in this form in two different ways. Can you find these 4 funny factorizations?
101. Four Consecutive Integers
Four Consecutive Numbers: a, b, c, d
Take any four consecutive numbers - for example 3, 4, 5, 6.
A. Why is (b x d) - (a x c) never even?
B. What is a + b + c - d always? Why?
C. What is (a2 + d2) - (b2 + c2) always? Why?
102. Two-Step Equation, Revisited
Source: Adapted from openmiddle.com/two-step-equations-2
Using the digits 1 to 9 at most one time each, place a digit in each box to create an equation in which the solution for x is 2 (so x = 2 is the answer to the equation). There are 32 different ways to do this - how many of them can you find?
103. Algebra Yohaku Puzzles
Source: yohaku.ca/algebraic-puzzles.html
Fill in each grid below with 9 different algebraic monomials so that each row and column has a product equal to the monomial expressions shown on the edges of the grid. For example, the product of the 3 monomials in the top row of the puzzle on the left must be a2b2c.
104. Patterns in an Infinite Table
Source: Teaching for Mathematical Understanding by Tony Cotton (submitted by Howie Hua, @howie_hua on TwiX)
Consider the table of counting numbers below. Assume the patterns shown continue infinitely. Some questions to consider:
105. Cryptogram
Source: brilliant.org
Each letter in the addition problem below represents a unique, single-digit number. Determine the values of A, B, C and D that make it a true sum.
106. Friedman Numbers
A Friedman number is a positive integer which can be written in some non-trivial way using its own digits, together with the symbols + – × / ^ ( ) and concatenation. Some examples:
25 = 52, 126 = 21 × 6, 216 = 62+1, 343 = (3 + 4)3, 688 = 8 × 86
Now determine why each of the numbers given below are Friedman numbers.
121 | 125 | 153 | 289 | 347 | 625 |
107. Happy Numbers
Source: wikipedia.org/wiki/Happy_number
On the left below is a demonstration of why 19 is a happy number. On the right, a demonstration of why 37 is NOT a happy number. Use these to determine what a happy number is then determine which of these numbers are happy and which are not: 13, 28, 44, 46 (then pick your own).
108. Visual Patterns
109. Even Steven
Source: artofproblemsolving.com
Below are the sums of the sequences of even numbers for n = 1 to 5 even numbers. Can you identify a pattern in the sums and use n to create a formula to express that sum? Try out your formula to find the sum of the first 10 even numbers, then the first 25.
n=1: 2 = 2
n=2: 2 + 4 = 6
n=3: 2 + 4 + 6 = 12
n=4: 2 + 4 + 6 + 8 = 20
n=5: 2 + 4 + 6 + 8 + 10 = 30
110. Reversi
Source: mathigon.org
When the number 9 is multiplied by the 4-digit number 1089, the result is the exact reverse of that number - 9801. There is also a 4-digit number that when multiplied by 4 will result in the reverse of itself. Find that 4-digit number!
9 x 1089 = 9801
4 x ABCD = DCBA
What is the 4-digit number ABCD?
111. What Numbers do you See?
Source: Simon Gregg (@Simon_Gregg) and JoAnn Sandford (@joann_sandford) on TwiX
)
Extension: What if the green triangle is equal to ¼?
112. Birthday Candles
Source: quora.com
)
Every year on my birthday, I have had a birthday cake with candles, one for each year of my birthday to date. In total, I have had 300 candles so far.
How old am I?
113. Men in Hats
Source: nrichmaths.org
)
Three men met for tea in their favorite cafe, taking off their hats as they arrived. When they left, they each put on one of the hats at random. What is the probability that they all left wearing the wrong hat?
What if there are four men?
114. Get the Digits
Source: @logicandmath on TwiX
)
Take a 4-digit number, then add the number formed by the first 3 digits to the number formed by the last 3 digits. For example:
1 2 3 4
123 + 234 = 357
If you do this and get 682, what 4-digit number did you start with?
115. Water Jugs
Source: wikipedia.org
)
You have an 8-gallon jug that is full of water. There is also 5-gallon jug and a 3-gallon jug that are both empty. Without using any other containers, how can you measure out exactly 4 gallons?
116. One Big Factor Family
Source: playwithyourmath.com
)
So there are four 1’s in 8’s family.
How many 1’s are there in 72’s family?
117. Number Cross 1 to 9
Source: @MathCirclesOZ on TwiX
)
Place the digits 1-9, once each, in the circles so that each crossing line has the same sum.
How many different solutions can you find?
118. Area vs Perimeter
Source: mathigon.org/puzzles#2017
)
119. 10 Cities and 5 Roads
Source: mathigon.org/puzzles#2017
)
120. 17 is a Prime Number
Source: Chris Smith’s Mathematics Newsletter, Issue #671, April 3, 2024 | @aap03102 on TwiX
)
How many numbers can you find with prime factors that add up to 17?
For example, the prime factorization of 80 is
2 x 2 x 2 x 2 x 5 and the sum of those factors is 2 + 2 + 2 + 2 + 5 = 13 (unfortunately, not 17). Organize your thoughts!
121. How Much for the Cookies?
Source: Dan Shuster | @DanShuster on Twix
)
Suppose that a bakery sells cookies in three different-sized packages: 2 for $2, 4 for $3 or 10 for $7. Depending on the available inventory, how much might a purchase of 24 cookies cost? What is the least and most you could spend on this purchase?
What about 36 cookies?
122. How Many Integer Points?
Source: Yacob Goitom | @ybgoi on Twix
)
How many integer coordinate points are on the blue line between point A and point B?
123. What is the Radius?
Source: Duane Habecker | @duanehabecker.bsky.social on BlueSky
)
This is a great problem for students in Geometry or any later course.
124. Six-Digit Equation
Source: AoPS Online | artofproblemsolving.com
)
125. A Basketball Puzzle
Source: Puzzle-a-Day Blog | puzzleaday.wordpress.com & @PuzzelADayBlog on TwiX
)
Amy played a game of basketball and scored a total of 19 points. In how many different ways could she have scored the 19 points? Note: In basketball, it’s possible to score points in three different ways: as one point (a free throw), as two points and as a three-pointer.
126. One Million
Source: Stella’s Stunners | mathstunners.org
)
Find two whole numbers whose product is 1,000,000 but neither of the two numbers has any zeros in it.
127. Peach Trees
Source: Stella’s Stunners | mathstunners.org
)
A rancher tells you that they have 10 peach trees, arranged in 5 straight rows with 4 trees in each row. What might this look like?
128. An Obscured Integer
Amy has written down five integers. These five integers have the unique property that their median is equal to their mean. When Belinda looks at the five integers, she sees 12, 17, 10, 21 and an obscured integer. What are the three possible values of the obscured integer?
129. A Puzzle From Survivor
This puzzle appeared on season 45 of the US TV show Survivor. Can you solve it?
130. A Sock Puzzle
Source: Puzzle-a-Day Blog | puzzleaday.wordpress.com
My sock drawer contains only black socks and white socks. If I pull two socks out of my drawer at random, then the chance of them being a black pair is a half, while the chance of them being a white pair is a twelfth. How many black socks, and white socks are in my drawer?
131. Mystery Dice
Source: Peter Williams | @MathsImpact on TwiX
I have a two blank 6-sided dice.
I write integers on each of their faces.
I roll both dice and add together their values.
These are the possible totals:
6 8 9 10 11 12 13 14 15 16 17
18 19 20 21 22 23 24 26 28 30
What numbers did I write on the dice?
132. The Billiard Ball Problem
Source: Dan Finkel | mathforlove.com/lesson/billiard-ball-problem & @MathforLove on TwiX
Suppose you launch a billiard ball from the lower left corner of the table at an angle of 45 degrees (see top right). Let the ball ricochet until it comes to a corner. Which corner will it end in? The 4x6 table at right ends in the upper left corner. Try different-sized tables and find any patterns you can about them.
133. Palindrome Numbers
Source: Dan Finkel | mathforlove.com/lesson/number-palindromes & @MathforLove on TwiX
A palindrome number is one that reads the same forward and backward, like 32523. There is a conjecture that you can turn any number into a palindrome number by doing the following:
Examples:
Try your own numbers - how many steps does it take?
134. “Squarable” Numbers
Source: Dan Finkel | mathforlove.com/lesson/squarable-numbers & @MathforLove on TwiX
In general, the number n is “squarable” if we can build a square out of precisely n smaller squares (of any size) with no leftover space. This is best demonstrated with an example. At right is one way to build a square from 11 smaller squares. Since we can cut a square into 11 smaller squares (of any size), we call the number 11 “squarable.” Grab some graph paper and see if you can find out which numbers from 1 to 25 are squarable!
135. 1 to 9 3-Digit Sum Puzzle
Source: New Zealand Maths | nzmaths.co.nz/resource/nine-tiles
Using the digits 1 to 9, with no repeats, fill in the boxes at right to make a true 3-digit sum. How many can you make? Can you see any patterns?
136. 1 to 9 Operations Puzzle
Source: Presh Talwakar | mindyourdecisions.com
Using the digits 1 to 9, with no repeats, fill in the boxes at right to make a true math operation. Note that it is a product combined with a sum.
137. Nine Squares
Source: Sara Carter | mathequalslove.net/nine-squares-puzzle & @mathequalslove on TwiX
Place the numbers 1 through 9 in the squares below in such a way that the number in any square in the upper row is equal to the sum of the numbers in the two squares immediately below it. How many solutions can you find?
138. Close to 1000
Using the digits 1 to 9 exactly one time each, fill in the boxes to make the sum as close to 1000 as possible. How close can you get? How many ways can you do it?
139. Seven Mystery Integers
Source: Puzzle a Day | puzzleaday.wordpress.com & @PuzzleADayBlog on TwiX
I have written down seven positive, ordered integers. The integers have a range of 6, a mean of 4, a mode of 1 and a median of 5 (the median only occurs once). What seven integers did I write down? Extension: create and share your own version of this puzzle.
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?
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140. The Tax Collector
Tax Collector is played like this: Start with a collection of paychecks, from $1 to $12. You can choose any paycheck to keep. Once you choose, the tax collector gets all paychecks remaining that are factors of the number you chose. The tax collector must receive payment after every move. If you have no moves that give the tax collector a paycheck, then the game is over and the tax collector gets all the remaining paychecks.
The goal is to beat the tax collector.
See the link below for examples, extensions
and further Information about this game.
141. Make Six
Source: Sarah Carter | mathequalslove.net/make-six-puzzle-number-challenge & @mathequalslove on TwiX
Insert mathematical symbols (no digits) to make each equation equal six. Hint: the square root symbol might come in handy in some cases!
142. Algebra/Geometry Triangle
Source: OCR Maths | @OCR_Maths on Twix
This problem is appropriate for students with a good amount of experience with algebra and geometry.
143. Twosday Challenge
Source: Sarah Carter | mathequalslove.net/twosday-challenge-activity & @mathequalslove on TwiX
Using exactly four twos, add arithmetical symbols between the twos to make each of the target numbers. You may use plus, minus, times, and divide symbols, as well as parentheses and brackets for grouping.
144. Quadrilateral w/Quadratic Curves
Source: OCR Maths | @OCR_Maths on Twix
This problem is appropriate for students with a good amount of experience with algebra and geometry.
145. Odd Sum Venn Diagram
Source: OCR Maths | @OCR_Maths on Twix
Be sure to use each digit exactly once each.
146. Product and Sum - 0 to 9
Source: OCR Maths | @OCR_Maths on Twix
Be sure to use each digit exactly once each.
147. Product and Sum - 0 to 9
Source: OCR Maths | @OCR_Maths on Twix
Be sure to use each digit exactly once each.
Replace the snowflakes with the digits 1-9, so that the number in each circle is the sum of the digits in the four adjacent squares.
148. Primes Sum to 100
Source: OCR Maths | @OCR_Maths on Twix
Six friends have a total of 100 dollars between them. Interestingly, the amount that each of them has individually is a prime number! How many dollars does each friend have?
How many different solutions can you find?
149. Two Calculations - 0 to 9
Source: OCR Maths | @OCR_Maths on Twix
Be sure to use each digit exactly once each.
150. Three Dice Sum
Source: OCR Maths | @OCR_Maths on Twix
151. Half-Hearted
Source: Transum | Transum.org & @Transum on Twix
Find the number which when added to both the top and bottom of each fraction makes the fraction equivalent to one half. Is there a pattern to this?
152. Broken Calculator
Source: Transum | Transum.org & @Transum on Twix
Use the keys on this broken calculator to make totals from one to ten. Five has already been done for you.
153. 1 to 8 Sums
Source: Transum | Transum.org & @Transum on Twix
154. Word Sums
Source: Transum | Transum.org & @Transum on Twix
Replace each letter with a different digit to make this word sum correct. Note that all F’s are the same digit, all T’s are the same digit, etc…
155. Four to Seven
Source: Transum | Transum.org & @Transum on Twix
156. Square and Even
Source: Transum | Transum.org & @Transum on Twix
Arrange the nine numbers on the squares so that each of the three digit numbers formed horizontally are square numbers and each of the three digit numbers formed vertically are even.
157. Happy New Year
Source: Transum | Transum.org & @Transum on Twix
What will the date be*:
*Modify this for any year as needed
158. One-Ninth
Source: Transum | Transum.org & @Transum on Twix
Three fractions add together to give one ninth.
159. Sum = Product
Source: Transum | Transum.org & @Transum on Twix
The sum of the numbers 1, 2 and 3 is the same as their product! (1 + 2 + 3 and 1 x 2 x 3 both equal 6)
160. Number Challenges
Source: Transum | Transum.org & @Transum on Twix
For each challenge, you will use all of the digits from 0 to 9 once each. Make 5 two-digit numbers…
161. Perimeter Expressions
Source: nrich Maths| nrich.maths.org/perimeterexpressions
Suppose you take a sheet of paper and cut it in half. Then cut one of those pieces in half, and repeat until you have five pieces altogether (see top right). Label the sides of the smallest rectangle, a for the shorter side and b for the longer side.
1. The shape at lower right can be made by combining the largest and smallest rectangles. Write an expression for its perimeter using a and b.
2. A different shape is made with the same two pieces and has a perimeter of 8a+6b. Can you figure out how make this shape?
162. Through the Window
Source: nrich Maths| nrich.maths.org/10344
The store in my town which sells windows calculates the price of windows according to the area of glass used and the length of frame needed.
Can you work out how they
arrived at the prices of the
windows at right?
Design another window and
Determine its cost.
380
140
150
310
350
340
163. Squares in Rectangles
A 2 by 3 rectangle contains 8 squares. Can you see how? A 4 by 6 rectangle contains 50 squares. Can you see how? | A 3 by 4 rectangle contains 20 squares. Can you see how? What size rectangle contains exactly 100 squares? Is there more than one? Can you find them all? Can you prove that there are no more? |
Source: nrich Maths| nrich.maths.org/4835
164. M, M and M
Source: nrich Maths | nrich.maths.org/mmandm
There are several sets of five positive whole numbers with the following properties:
Mean = 4, Median = 3, Mode = 3
Can you find all the different sets of five positive whole numbers that satisfy these conditions?
Can you explain how you know
you've found them all?
165. Friedman Numbers
Source: Erich Friedman | erich-friedman.github.io/mathmagic/0800
A Friedman number is a positive integer which can be written in some non-trivial way using its own digits, together with the symbols + – × / ^ ( ) and concatenation.
For example, 25 = 52 and 126 = 21 × 6.
Determine why each of these is a Friedman number.
25, 121, 125, 126, 127, 128, 153, 216, 289, 343, 347, 625, 688, 736, 1022, 1024,
1206, 1255, 1260, 1285, 1296, 1395, 1435, 1503, 1530, 1792, 1827, 2048, 2187, 2349,
2500, 2501, 2502, 2503, 2504, 2505, 2506, 2507, 2508, 2509, 2592 ,2737, 2916, 3125,
3159, 3281, 3375, 3378, 3685, 3784, 3864, 3972, 4088, 4096, 4106, 4167, 4536, 4624,
4628, 5120, 5776, 5832, 6144, 6145, 6455, 6880, 7928, 8092, 8192, 9025, 9216, 9261.
166. Area Patterns
Source: Mathshko | mathshko.com/2018/09/24/area-of-a-sector
167. The Painted Cube
Source: Credit to David Pimm | From Problems Worth Solving - Alicia Burdess
Picture a Rubik’s Cube. Now drop it into paint so that it is completely covered. When the paint is dry, imagine smashing it on the floor and breaking it apart into the smaller cubes.
How many of the cubes have one face covered in paint? How many cubes have two faces covered in paint? How many have three faces covered in paint? How many have zero faces covered in paint?
How could you predict the above for any size Rubik’s cube?
What about a 4 x 4 x 4? 5 x 5 x 5? 6 x 6 x 6? N x N x N?
168. Persistent Pattern?
Source: Michaela Epstein | @MathsCirclesOz on TwiX
169. 3 Questions
Source: Credit to The Journal of Economic Perspectives | In Problems Worth Solving - Alicia Burdess
170. Play this “Make 15” Game
Source: Credit to Peter Liljedahl & John Mason | From Problems Worth Solving - Alicia Burdess
Using the numbers 1 2 3 4 5 6 7 8 9
Alternate between partners to pick one number at a time.
Once a number is picked, it is gone. The goal is to have 3 numbers that add to 15.
What are some strategies for winning this game?
171. Same Sum
Source: Credit to Rina Zazkis | From Problems Worth Solving - Alicia Burdess
You have three cards below in front of you. On the back of each of the cards is a different prime number. The sum of the number on the front and the number on the back is the same for each card.
What are the prime numbers on the back of the cards?
Can you make your own
version of this puzzle?
172. Two Cubes Make a Calendar
Source: Puzzle-a-Day | puzzleaday.wordpress.com @PuzzleADayBlog on TwiX
An office worker has two cubes side-by-side on their desk. Each side of each cube has a digit on it (some digits may appear on both cubes). Every day the worker arranges both the cubes so that the front faces indicate the current day of the month, eg 2 and 8 for the 28th of the month. Keep in mind that single-digit numbers will have a leading 0 such as 05 for the 5th.
What digits are required on the faces
of the cubes so that all possible days
of the month can be expressed?
173. Two Cubes Make a Calendar
Source: Puzzle-a-Day | puzzleaday.wordpress.com @PuzzleADayBlog on TwiX
On each of the six lines in the image, the number at the center of each line should be the mean of the two numbers at each end. For example, if there is a 2 and a 6 at the end of a line, the center number will be a 4 since the mean of 2 and 6 is 4. Given the 3 numbers shown, your challenge is to fill in the remaining six numbers.
174. Difference of Two Squares
Source: nRich Maths | nrich.maths.org/whatspossible
Many numbers can be expressed as the difference of two perfect squares. For example,
20 = 62 − 42 21 = 52 − 22 36 = 62 − 02
Which of the numbers from 1 to 30 can you express as the difference of two perfect squares? What do you notice about the numbers you CANNOT express as the difference of two perfect squares?
175. Blue and White
Source: nRich Maths | nrich.maths.org/809
In the figures at right, identical squares of side one unit each contain some circles shaded in blue.
In which of the four figures is the shaded area greatest?
176. Take Three from Five
Source: nRich Maths | nrich.maths.org/1866
Select five random integers. Now see if you can find three of them that add up to a multiple of 3. Is this always possible, no matter which five numbers you start with? Explore this conjecture.
A starting point: choose 3 random integers. Is it always possible to select two of them that add up to an even sum (multiple of 2)?
177. Placing Sheets
Source: Math is Fun | mathsisfun.com/puzzles/placing-sheets.html
Eight squares of paper, all exactly the same size, have been placed on top of each other so that they overlap as shown.
In what order were the sheets placed?
178. A Weighty Problem
Source: Math is Fun | mathsisfun.com/puzzles/a-weighty-problem.html
I have ten boxes, with a total weight of 75kg:
15 kg, 13 kg, 11 kg, 10 kg, 9 kg, 8 kg, 4 kg, 2 kg, 2 kg, 1 kg
I want to pack the boxes into 3 crates, but each crate can only carry a maximum of 25 kg.
How can I pack the boxes into the crates?
How many different ways can this be done?
179. Nine Cannonballs
Source: Math is Fun | mathsisfun.com/puzzles/weighing-9-balls.html
You have 9 cannonballs, equally big, equally heavy - except for one, which is slightly heavier.
How would you identify the heavier cannonball if you could use a pair of balance scales only twice?
180. 1, 2, 3, 4 to make 1 through 10
Source: Math Walks | sites.google.com/powayusd.com/math-walks/math-walks-2020/june-2020
Make the numbers 1 through 10 using the numbers 1, 2, 3, 4 and +, −, x and/or ÷.
All the numbers 1, 2, 3, 4 must be used exactly once each for each number.
Any of the math operators can be used as many times as you like.