*** caution ***

Now, this Anniversary Coin has been stolen by someone.

***

Can you figure out whether the Anniversary coin is heavier or lighter using a balance scale at most six times?

]]>1. Weighing ABCDEFG vs. HIJKLMN

Case a) ABCDEFG = HIJKLMN This means that there a 6 or 8 (h) coins among the A-N, so O and P are from the same type

2. Weighing AB vs. OP

Case a) AB = OP This means all of ABOP are of the same kind. The third weighing is ABOP vs. HIJK. If ABOP is (l) then

A is (l). If ABOP is (h) then A is (h). A Balance ist not possible since then ABHIJKOP would be of the same kind, a

contradiction to the result of the first weighing

Case b) AB (l) OP. This means there is maximal one (h) coin among A and B. The third weighing is A vs B: If the weighing is

unequal, A’s status is settled. If it is balanced both A and B are (l)

Case c) AB (h) OP. This means there is maximal one (l) coin among A and B. The third weighing is A vs B: If the weighing is

unequal, A’s status is settled. If it is balanced both A and B are (h).

Case b) ABCDEFG (l) HIJKLMN In this case there are maximal 3 (h) coins among ABCDEFG

2. Weighing AB vs. CD

Case a) AB = CD This means there is maximal one (h) coin among A and B (otherwise we would have 4 (h) among ABCD).

The third weighing is A vs B: If the weighing is unequal, A’s status is settled. If it is balanced both A and B are (l)

Case b) AB (l) CD This means there is maximal one (h) coin among A and B. The third weighing is A vs B:

If the weighing is unequal, A’s status is settled. If it is balanced both A and B are (l)

Case c) AB (h) CD This means there is maximal one (l) coin among A and B

The third weighing is A vs B: If the weighing is unequal, A’s status is settled. If it is balanced both A and B are (h)

Case c) ABCDEFG (h) HIJKLMN just swap (h) and (l) in case b)

Test 1 Test 2 Test 3 Result Rationale

ABCDEFG < HIJKLMN AB < CD A ** B A heavier
A = B A lighter A=B lighter according to test 2
AB > CD symmetric case
AB = CD A **

A = B A lighter A=B=C=D all lighter according to test 1 and 2

ABCDEFG > HIJKLMN symmetric case

ABCDEFG = HIJKLMN A O A heavier

A = O AOP BCD A heavier A=O=P

AOP = BCD impossible A=O=P=B=C=D not possible according to test 1

Weighing Plan, where A is the Anniversary coin and B…O are the other coins.

Test 1 Test 2 Test 3 Result Rationale

ABCDEFG < HIJKLMN AB < CD A ** B A heavier
A = B A lighter A=B lighter according to test 2
AB > CD symmetric case
AB = CD A **

A = B A lighter A=B=C=D all lighter according to test 1 and 2

ABCDEFG > HIJKLMN symmetric case

ABCDEFG = HIJKLMN A O A heavier

A = O AOP BCD A heavier A=O=P

AOP = BCD impossible A=O=P=B=C=D not possible according to test 1