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Monthly Math Problem
Monthly Math Problem
Problem of the Month

January 2020 - Math Problem

You have a jar full of nickels, dimes and quarters.  If you select 3 coins, how many different sums of money are possible?  ​


​Solution to be posted at the end of the month!


December 2019 - Math Problem

Suppose you wanted to create a currency based on 4 coins.  What coin values would you use so that any amount between 1 and 10 could be created using at most two of these coins? (1, 2, 3, 7 is an example because 1, 2, 3, 2+2=4, 2+3=5, 3+3=6, 7, 7+1=8, 7+2=9, 7+3=10).  How many other examples are there? 


Solution to the problem from December

Suppose you wanted to create a currency based on 4 coins.  What coin values would you use so that any amount between 1 and 10 could be created using at most two of these coins? (1, 2, 3, 7 is an example because 1, 2, 3, 2+2=4, 2+3=5, 3+3=6, 7, 7+1=8, 7+2=9, 7+3=10).  How many other examples are there? ​


November 2019 - Math Problem

 

If there are three dots on each side of a hexagon as shown, then there are 12 dots in total.  

If there are five dots on each side of a hexagon as shown, then there are 24 dots in total.  


Following this pattern how many dots would there be in total if there were 10 dots on each side?  


Solution for November​​ 2019 Problem​​

There would be 54 dots.  Following this pattern, would it be possible to have a total of 85 dots?

October 2019 - Math Problem

If there were 8 people at a party and everyone shook everyone else’s hand exactly once, how many handshakes would there be in total?

Solution for October 2019 Problem​​


Yes.  2x3x4x5x6x7x8+1 or 40321 is one possible such number.  Are there others that are smaller than this one?


September 2019 - Math Problem


There are 50 tiles in a row.  Every third tile is red.  Every fourth tile has a star printed on it.  

Every remaining tile is green and unmarked.  

How many green and unmarked tiles are there?  

There are 26 green and unmarked tiles.  What if we started with 500 tiles instead of 50?  How many would be green and unmarked?  


Solution for ​September 2019 Problem​​

Score

Hits

3

3

6

3,3

7

7

9

3,3,3

10

3,7

12

3,3,3,3

13

3,3,7

14

7,7

15

Answer for 12 and 3

16

Answer for 13 and 3

17

Answer for 14 and 3

18 

Answer for 12 and 3 and 3


We can keep adding 3 to the answers for 12, 13 and 14 to get all of the numbers greater than 14.  This means that the highest score that we can not get is 11.  

What would be the highest score that you could not get if there were three sections on the dart board with the numbers 5, 7 and 9?

What would be the highest score that you could not get if there were two sections on the dart board with the numbers 2 and 4? 


June 2019 - Math Problem

Using all of the digits from 1 to 9 without repeating, make 3 three-digit numbers and add them up. How close to 1000 can you get without going over?

For example, one possibility would be 165+398+247=810.


Solution for June 2019 Problem​​

One way to get a sum of 999 is 537+168+294. Are there other ways to get a sum of 999? Is it possible to get a sum of 1000?​



May 2019 - Math Problem

Using the numbers 1, 3, 4 and 6, and the operations +, -, x, and / can you come up with the numbers from 1 to 20?  You must use the numbers 1, 3, 4 and 6 exactly once in each calculation. You may use brackets as part of your work.  For example:

1 =  4 x 1+ 3 - 6

2 =  4 + 3 - 6 + 1

3 = (6 + 3) / (4 - 1)


    Solution for May 2019 Problem​
    1 =  4 x 1+ 3 - 6

    2 =  4 + 3 - 6 + 1

    3 = (6 + 3) / (4 - 1)

    4 = (6 - 4) x (3 - 1)

    5 = 4 x 3 - 6 - 1

    6 = 6 - 4 +3 + 1

    7 = 4 x 3 - 6 +1

    8 = 6 + 4 - 3 +1

    9 = (6 - 3) x (4-1)

    10 = (4 + 1) x 6 / 3

    11 = (6 - 3) x 4 - 1

    12 = 6 + 4 + 3 - 1

    13 = 6 x 3 - 4 - 1

    14 = 1 + 3 + 4 + 6

    15 = 6 x 3 - 4 +1

    16 = 4 x (6 - 3 + 1)

    17 = 4 x 3 +6 -1

    18 = 4 x 3 + 6 x 1

    19 = 4 x 3 + 6 +1

    20 = 6 x 4 - 3 - 1

     

    Using 1, 3, 4, and 6 can you come up with 21, 22 and 23?  How about 24?   

    April 2019 - Math Problem

    There are 10 closed lockers in a hallway and 10 students. The first student walks down the hallway and opens every locker. The second student walks down the hallway and closes every second locker. The third student walks down the hallway and closes every third open locker and opens every third closed locker. This process continues for all 10 students. How many lockers are open at the end?


    Solution for April 2019 Problem

    Three (3) lockers will be open at the end.

      ​Locker 1 ​Locker 2 ​Locker 3 ​Locker 4 ​Locker 5 ​Locker 6 ​Locker 7 ​Locker 8 ​Locker 9 ​Locker 10
      ​Student 0 Closed Closed ​Closed ​Closed ​Closed ​Closed ​Closed ​Closed ​Closed ​Closed
      ​Student 1 Open ​Open ​Open ​Open ​Open ​Open ​Open ​Open Open​ ​Open
      ​Student 2 Open​ ​Closed ​Open ​Closed ​Open Closed ​Open ​Closed ​Open Closed​
      ​Student 3 ​Open ​Closed ​Closed ​Closed ​Open ​Open ​Open ​Closed ​Closed ​Closed
      ​Student 4 ​Open Closed​ ​Closed ​Open ​Open ​Open ​Open ​Open ​Closed Closed​
      ​Student 5 ​Open ​Closed ​Closed ​Open ​Closed ​Open ​Open ​Open ​Closed ​Open
      ​Student 6 ​Open ​Closed ​Closed ​Open ​Closed ​Closed ​Open ​Open ​Closed ​Open
      ​Student 7 ​Open ​Closed ​Closed ​Open ​Closed ​Closed ​Closed ​Open ​Closed ​Open
      ​Student 8 ​Open ​Closed ​Closed ​Open ​Closed Closed​ Closed​ ​Closed ​Closed ​Open
      ​Student 9 ​Open ​Closed ​Closed ​Open ​Closed ​Closed ​Closed ​Closed ​Open ​Open
      ​Student 10 ​Open Closed​ Closed​ ​Open ​Closed ​Closed ​Closed ​Closed ​Open ​Closed
       

      What if there were 16 lockers and 16 students at the start?  How many lockers would be open at the end?

      What if there were 400 lockers and 400 students at the start?  How many lockers would be open at the end?



      March 2019 - Math Problem

      In this​ sequence of numbers (3, 7, 10, 17, 27), after the first two, each number is the sum of the previous two numbers. If you want the fifth number to equal​ 100, what two positive numbers could you start with?


      Solution for March 2019 Problem​​​

      One possibility is 35 and 10 (35, 10, 45, 55, 100)

      Another is 20 and 20 (20, 20, 40, 60, 100)

      How many more can you come up with?


      February 2019 - Math Problem
       
      You have 4 different weights.  The sum of the weights is 40 grams.  If you have a two pan balance, you can use the 4 weights to make any weight from 1 gram to 40 grams.  How many grams is each of the weights?​


      Solution for February 2019 Problem​​​

      ​The weights are 1 gram, 3 grams, 9 grams and 27 grams.  

      Following this pattern, how many grams would the next weight be?  What range of weights could you make with these 5 weights?


        January 2019 - Math Problem

        You have 15 Loonies (one dollar coins) and four small bags. How many coins would you put into each bag so that you can pay any amount from​ $1 to $15 without opening bags?

        Solution for January 2019 Problem​​

        Bag Number of Coins
        ​A ​1
        ​B ​2
        ​C 4
        ​D 8
          
        Amount Bags
        ​$1 ​A
        ​$2 ​B
        ​$3 A, B
        ​$4 C
        $5 ​A, C
        ​$6 ​B, C
        $7​ ​A, B, C
        ​$8 ​D
        ​$9 ​A, D
        ​$10 ​B, D
        ​$11 ​A, B, D
        ​$12 ​C, D
        ​$13 ​A, C, D
        ​$14 B, C, D
        ​$15 ​A, B, C, D

        What if you had 31 coins and 5 bags?  How many coins would you put into each bag so that you can pay any amount from $1 to $31 without opening the bags?

        How many bags would​ you use if you had 127 coins and you wanted to pay any amount from $1 to $127?

         
        December 2018 - Math Problem

        This triangle has 4 dots on each side. A total of 9 dots were used to make the triangle.

        Image of triangle that has 4 dots on each side. A total of 9 dots were used to make the image of the triangle.

        The bottom triangle has 6 dots on each side. A total of 15 dots were used to make the triangle.

        Image of triangle that has 6 dots on each side. A total of 15 dots were used to make the image of the triangle.

        How many dots, in total, would be used to make a triangle with 10 dots on each side?

          Solution for December 2018 Problem​​

          One possible strategy: 10 dots on each side but 3 overlap, so there are 27 total dots.

          How many dots, in total would be used to make a triangle with 12 dots on each side? What about 100 on each side? Can you make a triangle in this way with a total of 56 dots? How do you know?

          November 2018 - Math Problem
          A game played with coins has the following rules.  

          1. Coins are set up in a ​triangle.  
          2. Coins are moved until the triangle is upside down.  
          3. The number of moves is recorded.  

          For example, if a two-layer triangle is used, then only one move is required to flip the triangle.  
          Image of a two-layer triangle where there is only one move is required to flip the triange


          A three-layer triangle would only require two moves.  
          Image of a three-layer triangle where only two moves are required

          How many moves would a triangle with 5 layers require?

            Solution for November 2018 Problem​​

            A 5 layer triangle requires 5 moves.

            Image: 5 layer triangle with letters A to O. 1st layer letter A; 2nd layer B & C; 3rd D, E, F; 4th G, H, I, J; 5th K, L, M, N,O               Image of triangle showing 5 moves. 1st layer letters K, D, E, F, O; 2nd layer G, H, I, J; 3rd L, M, N; 4th B & C; & 5th letter A


            October 2018 - Math Problem​


            If there were 8 people at a party and everyone shook everyone else's hand exactly once, how many handshakes would there be in total?

              Solution for October 2018 Problem​

              Number of People Who Have Arrived  ​Number of Handshakes
              ​0 ​0
              ​2 ​1
              ​3 3​
              ​4 6​
              ​5 10​
              ​6 15​
              ​7 ​21
              ​8 ​28

              How many handshakes would there be if there were 10 people? What about 100 people?
               

              September 2018 - Math Problem

               
              You throw a dart at the board below repeatedly. Each time you hit the board, you add either 7 or 3 to your score depending on which part you hit. What is the largest score that you can not get in this game? How do you know?


                 

                Solution for September 2018 Problem​​
                 
                Score 3     Hits 3
                Score 6     Hits 3,3
                Score 7     Hits 7
                Score 9     Hits 3,3,3
                Score 10   Hits 3,7
                Score 12   Hits ​3,3,3,3
                Score 13   Hits 3,3,7
                Score 14   Hits 7,7
                Score 15   Hits Answer for 12 and 3
                Score 16   Hits Answer for 13 and 3
                Score 17   Hits Answer for 14 and 3
                Score 18   Hits Answer for 12 and 3 and 3

                We can keep adding 3 to the answers for 12, 13, and 14 to get all of the numbers greater than 14. This means that the highest score that we can not get is 11.

                What would be the highest score that you could not get if there were three sections on the dart board with the numbers 5, 7 and 9?

                What would​ be the highest score that you could not get if there were two sections on the dart board with the numbers 2 and 4?


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