While playing one day, Megan decided to build cubes out of a box of individual sugar cubes. Megan emptied the box onto the floor and using the sugar cubes as building blocks, Megan made three solid cubes with no sugar cubes left over. Each of the cubes contained more than one sugar cube, and the three cubes may or may not have been the same size.
At this point, the family dog ran into the room and sent the cubes flying in all directions. The dog picked up one of the sugar cubes and left, crunching noisily.
Megan decided to start over, but this time she created two cubes instead of three. As before, each cube contained more than one sugar cube, the cubes may or may not have been the same size, and there were no sugar cubes left over.
What is the smallest number of sugar cubes that could have been in the box when Megan began?
h ttps://math.stackexchange.com/questions/3109484/puzzle-using-cubes
To find the smallest number of sugar cubes that could have been in the box when Megan began, let's examine the possibilities step by step.
For the first scenario when Megan made three solid cubes, we need to find the smallest number of sugar cubes that can be split evenly into three cubes without any left over.
We can start by listing some numbers that are divisible by 3:
3, 6, 9, 12, 15, 18, 21, 24, 27, 30...
However, the problem states that each of the cubes contained more than one sugar cube. So, we can eliminate the numbers 3 and 6 because the cubes would contain only one sugar cube each.
Let's continue with the remaining numbers:
9, 12, 15, 18, 21, 24, 27, 30...
At this point, we can see that the number 9 can be used to build three cubes, each containing 3 sugar cubes. So, the smallest number of sugar cubes in this case is 9.
Now, let's move on to the second scenario where Megan created two cubes instead of three. We need to find the smallest number of sugar cubes that can be split evenly into two cubes without any left over.
We can list some numbers divisible by 2:
2, 4, 6, 8, 10, 12, 14, 16, 18, 20...
Again, we need to take into account that each cube should contain more than one sugar cube. Therefore, we can eliminate the numbers 2, 4, 6, and 8.
Continuing with the remaining numbers:
10, 12, 14, 16, 18, 20...
We can see that the number 12 can be used to build two cubes, each containing 6 sugar cubes. So, the smallest number of sugar cubes in this case is 12.
Comparing the two scenarios, the smallest number of sugar cubes that could have been in the box when Megan began is the smaller of the two numbers: 9.
Therefore, the smallest number of sugar cubes is 9.