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Depth · one concept from first principles · step 2 of 4

Run a counting rule backward to every input

Each natural number x is paired with how many positive even numbers are at most x. Find every x that gets paired with 5, and add them.
ExpressionsOperations
Your answer
How to solve
Strategy Look for a Pattern — List the count as x grows and watch the pattern: y climbs by 1 at each even number and stays put on the odd number after it. The values of x with y=5 form a short stretch, which we list and add.
1STEP 1

List the count near y=5

The positive even numbers in order are 2, 4, 6, 8, 10, 12, … Up to 9 there are four of them; 10 brings the fifth; 12 brings the sixth.

x = 9 → y = 4, x = 10 → y = 5, x = 11 → y = 5, x = 12 → y = 6
2STEP 2

Find the stretch where y=5

Exactly five evens, 2, 4, 6, 8, 10, are at most a when a has reached 10 but not yet 12.

10 ≤ a ＜ 12
3STEP 3

Add the possible values

The natural numbers in that stretch are 10 and 11.

10 + 11 = 21
Answer
21
For a=10 the evens are 2, 4, 6, 8, 10, which is five; for a=11 they are the same five; a=9 gives four and a=12 gives six. So exactly 10 and 11 work.
Takeaway

When a rule counts things, many inputs can share an output -- find the whole stretch, not just one value.

  • List the count near y=5
  • Find the stretch where y=5
  • Add the possible values
Where next?
Level upsuggested
Exactly one prime sum makes a functionTwo equal inputs break a function
Another one like thissuggested
12 practice problems
Step back and solidifysuggested
Run the sum rule backwards on unit fractionsGroup by numerator plus denominator, then count

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▶ Practice — 12 problems