Last week’s Micro Softy featured three bags of marbles. The surprising solution to that puzzle is below. But first, here’s today’s brain tickler, beginning with some background on it:

The original twin paradox is part of Einstein’s theory of relativity: A twin traveling at high speed in space ages more slowly than the twin who stays on Earth, due to time dilation. The paradox shows that time travel into the future is theoretically possible.

Here’s another less profound, Earthbound twin paradox: Curt and Rod were brothers in business together in a 24-7 store that sold drapes. They alternated 12-hour shifts working at the store. To customers, it looked like the same man was always behind the counter because Curt and Rod looked alike. In fact, Curt and Rod were born on the same day of the same year and had the same mother.  

Yet Curt and Rod were not twins. Can you resolve this twin paradox?

Solution to Micro Softy 17:    Bags of Marbles

First here’s the puzzle: There are three bags, each containing 100 marbles. One bag contains only black marbles, another contains only white ones, and the third contains a mix of both black and white marbles. All three bags are labeled but labeled incorrectly. The challenge is: How many marbles do you need to draw, and from which bag(s), to correctly identify the contents of all three bags?

The answer, surprisingly, is one marble. As shown in the Figure, the single marble should be drawn from the bag marked MIXED.

The MIXED bag, like the other two, is labeled incorrectly. So, if the single marble drawn is white, the bag should be relabeled WHITE. Now only the MIXED and BLACK labels are left to be reassigned between the two remaining mislabeled bags. Because the BLACK bag is wrongly labeled, it must actually be MIXED. Thus the final bag contains all black marbles and should be labeled BLACK. This outcome is illustrated in the second row of the Figure.

Similar reasoning applies if the single marble drawn from the MIXED bag is black. As shown in the third row of the Figure, the bag originally marked MIXED is now relabeled BLACK. This leaves incorrectly labeled bags  WHITE and MIXED. Because the bag marked WHITE is wrongly labeled, its correct label must be MIXED and the remaining bag’s label must be WHITE.

All this information from a single marble! That’s the power of logic.

Also, here’s a short vid on Einstein’s Twin Paradox:

Here are the puzzles (and solutions) from the last six Mondays:

Monday Micro Softy 12: Can You Connect the Dots? You may use no more than four perfectly straight lines and the lines must be connected. This classic puzzle in simple graph theory resulted in a commonly used phrase. Can you guess it?

Monday Micro Softy 13: Garbage Trucks, String Theory… … and Stained-Glass Windows. What connects them? These puzzles, associated with the great mathematician Leonard Euler (1707‒1783), have a practical use, for example, in laying water pipes.

Monday Micro Softy 14: How Did the Blind Ticket Seller Know? This puzzle doesn’t require math skills so much as advanced common sense reasoning. About last week’s solution, given here: If you code, the second part of the puzzle could also be offered to a computer.

Monday Micro Softy 15: What Happens to the Hole in a Hot Washer? When a washer ring is heated, does the hole in the center get bigger or smaller? The answer to last week’s Micro Softy turns on the question of what form of currency Claire gave the blind ticket seller. Did you guess it right?

Monday Microsofty 16: The Leaky Bucket. The puzzler must decide where to place the hole for maximum distance of outward flow. And explain how the problem would change on the Moon. One hint for the solution to the hot washer problem, given here, is the old trick for getting a tight lid off a jar: Running hot water over it.

Monday Micro Softy 17: Mixed-Up Bags of Marbles: The bags-of-marbles puzzle is comparatively simple: How many marbles must you pull from the mislabeled bags in order to relabel them correctly? There’s a rough solution to the “leaky bucket” problem but an exact solution requires some use of fluid dynamics. But the Moon, surprisingly, changes nothing.

Note: At Monday Micro Softy 11: What Happened to That Other Dollar?, you will also find links to the first ten Micro Softies. Have fun!