You are a detective famous across the vast reaches of Chippewa county, Michigan. You just intercepted a shipment of 6561 croissants that originated in Abitibi, Quebec.
You are immediately suspicious because Abitibi is home to your nemesis the feared French-Candien gang Trois-Fils. Rumor in the swamps is that the Trois-Fils are the source of hopium, the new, synthetic heroin that is destroying the country.
Hopium is odorless, does not show up on x-rays and is otherwise undetectable except for the fact that hopium has approximately the same mass-density as gold. Hopium is also astronomically potent. The amount of hopium the size of a croissant is enough to turn every tweaker in the United States into a mindless zombie.
To nobody's surprise, the shell corporation that owns the shipment of croissants files an injunction with the 9th District Court to prevent you from inspecting the shipment. To everybody's surprise the 9th District Court will allow you to test the chemistry of one sample croissant.
The Chippewa county District Attorney files a counter-suit and the 9th District Court amends their decision. They will allow you to make 8 weight measurements before you choose your sample to send to the lab.
Can you find the Hopium Croissant with only eight, weight measurements?
You can if you exercise some creativity when interpreting the phrase 'weight measurements'. You don't weigh the croissants against a known weight standard, you compare their weights against each other.
You take your balance beam scale into the evidence warehouse.
You separate the 6561 croissants into three equal piles of 2187 croissants.
You place one lot of 2187 on one tray of your balance beam and another lot of 2187 on the other. If one lot is heavier than the other then you know the heavy lot holds the Hopium Croissant. If the two lots are equal then you know the Hopium Croissant is in the lot you did not put on the balance beam.
Further divide the suspect lot of 2187 croissants into lots of 729 for the second measurement. Repeat as before.
The third measurement will be lots of 243.
The fourth measurement will be lots of 81.
The fifth measurement will be lots of 27.
The sixth measurement will be lots of 9.
The seventh measurement will be lots of 3.
The eight measurement will be lots of 1.
The moral of this story is that even "failed" tests contain information. The inability to distinguish a weight difference between two lots of croissants contains critical information...if you know how to use it.