BUILDING THINKING
CLASSROOMS
Good Problems
Three cards (see below) each have a prime number on the back. The sum of the number on the front and back of each card is the same as the other cards. What are the numbers on the back of each card?
Lost Primes
Imagine a typical 6sided die, and notice that the sum of opposite faces is always seven. The one is across from the six, the two is across from the five, and so on. Now imagine that you were making your own sixsided die that did not have this restriction. How many different dice could be made?
Scrambled Dice
Paint all the sides of a 3 x 3 x 3 cube. Once it is dry take it apart into its 1 x 1 x 1 unit cubes. How many of these unit cubes have paint on three faces? Two faces? One face? No faces? Explore for 4 x 4 x 4, etc.
Painted Cube
How many squares on a chessboard?
How many rectangles?
Chessboard
Consider the string 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Cross out any two numbers in this list and add the difference to the end of the list. This new number is now part of the list. Continue the process of crossing out two number on the list and adding the difference until there remains only one number. What can you say about the last number? Explore. [from Richard Hoshino]
The Last Number
Consider the number 28. It is not a palindrome. So, I reverse the number and add it to itself (28 + 82 = 110). 110 is not a palindrome. So, I reverse it and add it to itself (110 + 011 = 121). 121 is a palindrome. This means that 28 is a depth 2 palindrome (it took two iterations to make it a palindrome).Â Â What is the depth of all two digit numbers?
Palindromes
Consider the following four dice and the numbers on their faces:

Red : 0, 1, 7, 8, 8, 9

Blue: 5, 5, 6, 6, 7, 7

Green: 1, 2, 3, 9, 10, 11

Black: 3, 4, 4, 5, 11, 12
These are used to play a game for two people. Player 1 chooses one of the die to use for the game. Then player 2 chooses a die. Now each player rolls their die. The player with the highest number showing gets a point. The first player to 7 points wins the game. If you are player 1 which die should you choose. If you are player 2 which die should you choose? [from the Mathematics Task Centre]
Duelling Dice
How do you make a 9 minute egg if all you have is a 4 minute and 7 minute egg timer?
Egg Timer
You have 27 small cubes, 3 each of nine colours. Use the small cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of every colour. [from nRICH]
Rubic’s Cubeish
What is a better fit, a square peg in a round hole or a round peg in a square hole?
Square Peg in a Round Hole
Honeybees have a very interesting genealogy. While female honeybees come from a fertilized egg, male honeybees come from an unfertilized egg. That is, female honeybees have a mother and a father, while male honeybees have only a mother. Look at the family tree of a male honeybee. How many ancestors are there in the previous generation? The one before that? The one before that? How many ancestors 10 generations ago? Etc.
Honeybee Ancestors
Consider the differentiable function below. What are m and b?
Differentiable Function
How many 6 digit numbers are there whose digits sum to 51? [from Practice Fermat Number 4, #7]
Sum of 51
On Canada Day, Alex stacks dice on top of each other until the sum of the visible faces are equal to the year. When will he next be able to do this? Generalize.
Canada Day
If you write out the numbers from 1 to 1000, how many times will you write the number 7?
How Many 7’s
Make the numbers from 1 to 30 using four 4’s and any operations.
Fantastic Four
In how many ways can 105 be expressed as the sum of at least two consecutive positive integers? [from the Canadian Team Math Contest, 2014, #12]
105
On the desk calendar, what numbers have to be where in order for every date from 01 to 31 to be able to be represented?
The Desk Calendar
Determine the minimum perimeter of a triangle with one vertex at (7,1), one vertex on the xaxis, and one vertex on the line y = x. [from Richard Hoshino]
Triangle Perimeter
There are 1001 pennies lined up on a table. I come along and replace every second coin with a nickel. Then I replace every third coin with a dime. Finally, I replace every fourth coin with a quarter. How much money is on the table?
1001 Pennies
Two darts are thrown at a circular target. Where does the first dart have to land so that the probability that the second dart will be closer to the centre is 50%?
Dart Board Probabilities
Adam, Bob, Clair and Dave are out walking: They come to rickety old wooden bridge. The bridge is weak and only able to carry the weight of two of them at a time. Because they are in a rush and the light is fading they must cross in the minimum time possible and must carry a torch (flashlight,) on each crossing.
They only have one torch and it can’t be thrown. Because of their different fitness levels and some minor injuries they can all cross at different speeds. Adam can cross in 1 minute, Bob in 2 minutes, Clair in 5 minutes and Dave in 10 minutes.
Adam, the brains of the group thinks for a moment and declares that the crossing can be completed in 17 minutes. There is no trick. How is this done? [from Nigel Coldwell]
Crossing the Bridge
Suppose there was one of six prizes inside your favorite box of cereal. Perhaps it’s a pen, a plastic movie character, or a picture card. How many boxes of cereal would you expect to have to buy, to get all six prizes? [from George Reese]
Cereal Prizes
You have two ropes, each of which takes two hours to burn if lit at one end. These ropes are not homogeneous and some parts of the ropes burn quicker than others. Can you use the ropes to time out 1 hour and 30 minutes? 45 minutes? What other times could you get? [from TestFunda, book 1, #7]
Burning Rope
Consider two whole numbers (for example 3 & 6). These will be the first two numbers. The third number is the sum of the first two (9). The forth is the sum of the previous two (15), and so on (3, 6, 9, 15, 24, 39, …). What do the first two number have to be such that the fifth number is 100?
Seed Numbers
Imagine a series of equally spaced holes running along a line infinitely in both directions. A groundhog is in one of the holes and every minute he jumps to a new hole that is some fixed interval of holes away from his current hole. You do not know where he starts, or the interval that he jumps, or the direction that he goes in, but after each of his moves you can shine a flashlight into one (and only one) hole. The problem is to find a method by which you can guarantee to eventually find the groundhog.
Groundhog Hunt
Students in a marching band want to line up for their performance. The problem is that when they line up in 2s there is 1 left over. When they line up in 3s there are 2 left over. When they line up in 4s there are 3 left over. When they line up in 5s there are 4 left over. When they line up in 6s there are 5 left over. When they line up in 7s there are no students left over. How many students are there? [from John Grant McLoughlin]
Marching Band
We place in a box thirteen white marbles and fifteen black. We also have twentyeight black marbles outside the box. We remove two marbles from the box. If they have a different colour we put the white one back in the box. If they have the same colour we put a black marble in the box. We continue doing this until only one marble is left in the box. What is its colour? [from Vector 55(1) – p. 49]
Black and White Marbles
Three jugs have the capacity for 8, 5 and 3 litres respectively. The 8 litre jug is filled entirely with water and the other two jugs are empty. Your task is, by decanting, to divide the water into two equal parts, that is, 4 litres in jug A and 4 litres in jug B, leaving the smallest jug empty. [from MathXTC – Neurological Nasties (problem set 9, #3)]
Three Jugs
An escaped prisoner finds himself in the middle of a square swimming pool. The guard that is chasing him is at one of the corners of the pool. The guard can run faster than the prisoner can swim. The prisoner can run faster than the guard can run. The guard does not swim. Which directionÂ should the prisoner swim in in order to maximize the likelihood that he will get away? [from Vector 54(2) – p. 95]
Escaped Prisoner
Four Aces: Figure out this trick
Determine the minimum perimeter of a triangle with one vertex at (7,1), one vertex on the xaxis, and one vertex on the line y = x. [from Richard Hoshino]
Triangle Perimeter
There are 1001 pennies lined up on a table. I come along and replace every second coin with a nickel. Then I replace every third coin with a dime. Finally, I replace every fourth coin with a quarter. How much money is on the table?
1001 Pennies
Two darts are thrown at a circular target. Where does the first dart have to land so that the probability that the second dart will be closer to the centre is 50%?
Dart Board Probabilities
We know that the sum of exterior angles of any convex polygon is 360. But what if the polygon is concave? What adjustments or extension to the ‘rule’ must be made in order for the sum to still be 360? [from Robert McGregor on the AAMT]
Concave Polygons
When 100 is divided by a positive integer x, the remainder is 10.
When 1000 is divided by x, what will the remainder be? [from University of Waterloo CECM Past Contest Archive (2011 Caley, #18)]
100 Divided
10 Prisoner Release
A warden has decided that his jail is too full, so he wants to release some of the prisoners. His strategy for doing this is that he will take ten prisoners at random and line them up in a row so that each prisoner can only see the prisoners in front of them. The warden will then place a black or white hat on the head of each prisoner. Starting at the back of the row the warden will then ask that prisoner what colour his hat is. If he guesses correctly then he is released. The warden then goes to the next prisoner in line, and so on. The ten prisoners are allowed to meet to discuss before they line up. If they are clever they will come up with a strategy that will guarantee to save some of them. How many could you save?