Depth · one concept from first principles · step 2 of 4
Fix one unknown and count the rest
Count every ordered triple of natural numbers (x, y, z) whose sum is 7. Order matters, so (1, 2, 4) and (4, 2, 1) are different.
Expressions
Your answer
How to solve
Strategy Make a Systematic List — Three unknowns at once is hard to list, so split the problem by the value of x. Each case leaves a two-unknown equation y+z=n, whose count follows a simple pattern. A systematic list of the cases then gives the total.
1STEP 1
Find the possible values of x
Since y and z are each at least 1, y+z is at least 2, so x is at most 7-2=5.
x ∈ {1, 2, 3, 4, 5}
Fixing one unknown turns one hard count into several easy ones.
6.EE.B.5Identify Subproblems2STEP 2
Count the solutions of y+z=n
For y+z=n with natural numbers, y can be 1, 2, …, n-1 and then z is decided, so there are n-1 pairs. With n=7-x that is 6-x pairs.
y+z=6: (1,5),(2,4),(3,3),(4,2),(5,1) → 5 pairs
The pattern n-1 is visible from one small case.
7.EE.B.4Look For A PatternFor a fixed , the equation has exactly pairs of natural numbers.
Why?
Once is chosen, has no freedom left: it must be , because taking away undoes adding it.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Why?
is at least only when is at most , so each of matches exactly one pair, and the pairs are as many as those numbers.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
3STEP 3
Add the cases
List the counts for x=1 to 5: they are 5, 4, 3, 2, 1.
5 + 4 + 3 + 2 + 1 = 15
Each case is separate, so the counts simply add.
6.EE.B.5Make A Systematic ListAnswer
15
The largest any one unknown can be is 5 (the other two at least 1 each), and the counts shrink by one as x grows, which is what the list 5, 4, 3, 2, 1 shows. A total of 15 is reasonable for such a small sum.
Takeaway
Hold one unknown still, count what is left, and add up the cases.
- Find the possible values of x
- Count the solutions of y+z=n
- Add the cases
Where next?
Another one like thissuggested
Step back and solidifysuggested
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The same shape shows up later on the AMC, as Arithmetic Expression Decomposition, Branching State Enumeration, Small-Domain Constrained Search. Why the shapes repeat →