Competition · AMC preparation · step 4 of 4
AMC 8 · 2000 · #8
Grade 3 arithmeticgeometry-3d
Pick an answer.
AMC 8 2000 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
We are never told which face of each die points where, and we cannot see eleven of them — so trying to identify each hidden face is the wrong path. The invariant (Tool #11) is the sum of the six faces on a single die: 1+2+3+4+5+6 = 21 no matter how the die is turned. Tool #9 (Solve an Easier Problem) tells us to first solve for one die, then scale to three. Once we know the grand total of dots on all three dice, the hidden total is just total minus visible — a single subtraction.
Add the dots on one die
One die's six faces sum to 21, and this total never changes when you turn the die.
This is the invariant. Whichever face is up, down, front, back, left, or right, the six numbers still add to 21.
3.OA.D.8Solve An Easier Related ProblemScale up to three dice
Three dice give 3 × 21 = 63 dots across all eighteen faces.
Three identical groups of 21 — a Grade 3 multiplication.
3.OA.A.1Work BackwardsAdd the visible dots
The seven visible faces 1, 1, 2, 3, 4, 5, 6 add up to 22.
A straight addition of seven small numbers.
3.NBT.A.2Solve An Easier Related ProblemSubtract visible from total
Every face is seen or hidden, so hidden dots = 63 − 22 = 41.
Everything on the dice is either seen or hidden, so the two parts must add back to the whole. Subtracting the seen part leaves the hidden part.
The number of dots on the eleven hidden faces equals the total dots on all three dice minus the dots on the seven visible faces.
▸ Why?
Every one of the eighteen faces is either seen or hidden, with none left out and none counted twice, so the visible dots and the hidden dots together make the whole total.
▸ Why?
Since the visible part and the hidden part add up to the total, taking the visible dots away from the total leaves exactly the hidden dots.
▸ Why?
The total the two parts add up to is a fixed 63 that does not change with how the dice are turned.
▸ Why?
Each single die always carries the numbers 1 through 6, and turning a die only changes which face points where, not the collection of numbers, so reordering them never changes their sum of 21.
▸ Why?
Three dice are three equal groups of 21 dots, and three groups of 21 is 3 × 21 = 63.
The six faces of a die always add to 21, no matter how it is turned. Three dice hold 63 dots in all, so the hidden dots are just 63 - 22 = 41 — one subtraction once you spot the invariant.
- Add the dots on one die
- Scale up to three dice
- Add the visible dots
- Subtract visible from total
A parent dashboard for the family lives at sensimlab.com.