Problem
Find the lighter counterfeit coin
Three look-alike coins contain exactly one counterfeit. The fake is lighter than a real coin. The balance may be used only once. Describe how to find the fake.
Reasoning · Grade 6-2 Balance Scales and Counterfeit Coins
Your answer
Watch out The answer seems to be on the pans, but the coin left off can be it.
How to solve
Strategy Make a Systematic List — There are only three suspects, and one weighing gives only three answers. Those two threes are the whole problem, so I list the three things the balance can do and check that each one points at a different coin. That is also what tells me how to load the pans: one coin on each side and the third coin left on the table. Keeping a coin off the scale is what turns the 'beam stays level' outcome — which would otherwise be wasted — into a useful signal.
1STEP 1
Count what one weighing is able to say
One weighing can say three things.
3 possible outcomes ≥ 3 suspect coins
Playing with a balance is a Kindergarten activity: it tips one way, tips the other way, or stays flat. Counting those three possibilities is the whole 'clever' idea in this problem.
K.MD.A.2Make A Systematic ListOne weighing can say only three things, so it can separate at most three possibilities.
Why?
The beam tips left, tips right, or stays level, and those three outcomes never overlap and leave nothing out.
🧱Disjoint cases addSplit the possibilities so no case overlaps another and none is left out, and the case counts simply add.
Why?
If there were more possibilities than outcomes, two of them would have to share an outcome and could never be told apart.
🧱More items than boxes forces a shared boxn boxes can hold at most n items one apiece, so item number n plus one has nowhere new to go and must share.
2STEP 2
Choose the weighing: one coin on each pan, one coin left off
One coin per pan and one left off.
left pan = ① right pan = ② off the scale = ③
Draw the three coins in a row and circle the two that go on the pans. The circle you did not draw is the coin the level beam will point at — the picture makes the plan easy to remember.
K.MD.A.2Draw A Diagram3STEP 3
Outcome ① ＞ ②: the right pan rose, so ② is the counterfeit
The pan that rises holds the fake.
① ＞ ② → ② is the counterfeit
Watch out for the direction: the fake is lighter, so its pan goes UP. The coin to accuse is the one on the higher pan, not the lower one.
K.MD.A.2Eliminate Possibilities4STEP 4
Outcome ① = ②: neither weighed coin is fake, so ③ is
If level, the coin left off, 3, is the fake.
① = ② → ①, ② genuine → ③ is the counterfeit
This is the Grade 1 idea of comparing two things indirectly through a third. I never weighed ③ against anything, but learning that the other two are equal, plus knowing that exactly one coin is light, compares ③ with them for me.
1.MD.A.1Eliminate Possibilities5STEP 5
Outcome ① ＜ ②: the left pan rose, so ① is the counterfeit
Tipping the other way names the other coin.
① ＜ ② → ① is the counterfeit
Once you have argued one tipping case, the other one needs no new thinking — swapping the words 'left' and 'right' turns one argument into the other.
K.MD.A.2Eliminate Possibilities6STEP 6
Check the rule by running it backwards
The three outcomes match the three coins one to one.
① ＞ ② → ② ① = ② → ③ ① ＜ ② → ①
Checking a plan forwards ('what does this signal mean?') can hide a gap; checking it backwards ('what signal would this truth make?') catches any case the plan forgot.
K.MD.A.2Make A Systematic ListAnswer
weigh 1 against 2, leaving 3 off
3 = 3
The rule must cover every possibility and never give two answers at once. One weighing has exactly 3 outcomes and there are exactly 3 suspects, and the plan pairs them up one-to-one: ① ＞ ② goes to ②, ① = ② goes to ③, ① ＜ ② goes to ①. No outcome is left without a meaning and no coin is left unaccused, so nothing is wasted and nothing is double-booked. The direction also makes sense: the counterfeit is lighter, so the pan holding it must rise, which is why the answer is always the coin on the higher pan — never the lower one. And running the plan backwards from each of the three possible truths reproduces a different signal each time, which is exactly what a one-weighing method needs.
Takeaway
A balance can answer in three ways, so weigh only two of the three coins and let 'stays level' be the third coin's answer.
- Count what one weighing is able to say
- Choose the weighing: one coin on each pan, one coin left off
- Outcome ① ＞ ②: the right pan rose, so ② is the counterfeit
- Outcome ① = ②: neither weighed coin is fake, so ③ is
- Outcome ① ＜ ②: the left pan rose, so ① is the counterfeit
- Check the rule by running it backwards