Reasoning · Grade 6-2 Balance Scales and Counterfeit Coins

Problem

Find a counterfeit of unknown weight

Four look-alike coins ①②③④ contain exactly one counterfeit, and nobody says whether the fake is lighter or heavier. The balance may be used twice, and a table holds one row per pair of outcomes. Two rows are filled in already: ① balancing ② then ① balancing ③ points at ④, and ① balancing ② then ① outweighing ③ points at ③. Fill in the rest of the table.
Weighing 1 Weighing 2 Counterfeit coin 1 = 2 1 > 2 1 < 2 1 = 3 4 1 > 3 3
Your answer
How to solve
Strategy Make a Systematic List — The table itself dictates the method: walk the three branches of weighing 1 one at a time, and inside each branch walk the three results of weighing 2. What makes this harder than the version where the fake is known to be lighter is that every coin now carries two suspicions — 'light fake' and 'heavy fake' — so there are 4 x 2 = 8 situations to separate instead of 4. The single fact that keeps the casework short is that a level beam certifies its coins as genuine, which turns ① into a trusted ruler for the second weighing. Two of the nine rows turn out to be impossible, and saying why is part of the answer.
1STEP 1

What the balance can say, and how many rows that makes

Two weighings make a table of nine rows.

3 results × 3 results = 9 rows
2STEP 2

Count the situations the table has to separate

There are eight situations, so nine rows suffice.

4 coins × 2 directions = 8 situations ＜ 9 rows
3STEP 3

The fact that does all the work: a level beam certifies genuine coins

A level beam means those coins are genuine.

① = ② → ① and ② are both genuine
4STEP 4

Block 1 — weighing 1 gives ① = ②: the fake is ③ or ④

If the first is level, the fake is 3 or 4.

① = ②: ① = ③ → ④, ① ＞ ③ → ③ (lighter), ① ＜ ③ → ③ (heavier)
5STEP 5

Block 2 — weighing 1 gives ① ＞ ②: the fake is ① heavy or ② light

If it tips, the fake is one of those two.

① ＞ ②: ① = ③ → ② (lighter), ① ＞ ③ → ① (heavier), ① ＜ ③ → impossible
6STEP 6

Block 3 — weighing 1 gives ① ＜ ②: the mirror image of block 2

Tipping the other way is the mirror image.

① ＜ ②: ① = ③ → ② (heavier), ① ＜ ③ → ① (lighter), ① ＞ ③ → impossible
7STEP 7

Check the finished table against all 8 situations

All eight situations land on their own row.

8 situations → 7 reachable rows + 2 impossible rows = 9
Answer
weigh 1 against 2, then 1 against 3
4 × 2 = 8
There are 4 x 2 = 8 possible situations and 3 x 3 = 9 rows, so a two-weighing table is not obviously doomed — and indeed 7 of the 9 rows are reachable and cover all 8 situations, with the single doubled-up row holding two situations about the same coin ④. Every reachable row ends in exactly one coin, so no branch leaves me guessing, which is the real test of this table. The two blank rows are not an oversight: each of them would require three coins with three different weights, which contradicts 'exactly one coin is off-weight'. Directions also check out — in every tipping branch the coin accused sits on the side the beam predicts (heavy fake on the lower pan, light fake on the higher pan). Finally, the table shows why not knowing the direction genuinely costs something: with a known-lighter fake, four coins could be handled with two weighings and plenty of room to spare, whereas here the eight situations only just fit.
Takeaway

When you do not know if the fake is light or heavy, every coin becomes two suspects — so use the first weighing to make a coin you can trust, then measure the suspects against it.

  • What the balance can say, and how many rows that makes
  • Count the situations the table has to separate
  • The fact that does all the work: a level beam certifies genuine coins
  • Block 1 — weighing 1 gives ① = ②: the fake is ③ or ④
  • Block 2 — weighing 1 gives ① ＞ ②: the fake is ① heavy or ② light
  • Block 3 — weighing 1 gives ① ＜ ②: the mirror image of block 2
  • Check the finished table against all 8 situations