Did you enjoy Brain Teaser Week, fellow puzzlers and PuzzleNationers? We certainly hope so! It was a fun experiment in dedicating an entire week to a particular type of puzzle.
We gave you three puzzles to challenge your deductive, mathematical, and puzzly skills, and now it’s time to break them down and explain them.
A set of football games is to be organized in a “round-robin” fashion, i.e., every participating team plays a match against every other team once and only once.
If 105 matches in total are played, how many teams participated?
If every team plays every other team once, you can easily begin charting the matches and keeping count. With 2 teams (Team A and Team B), there’s 1 match: AB. With 3 teams (A, B, and C), there are 3 matches: AB, AC, BC. With 4 teams (A, B, C, and D), there are 6 matches: AB, AC, AD, BC, BD, CD. With 5 teams (A, B, C, D, and E), there are 10 matches: AB, AC, AD, AE, BC, BD, BE, CD, CE, DE.
Now, we could continue onward, writing out all the matches until we reach 105, but if you notice, a pattern is forming. With every team added, the number of potential matches increases by one.
With one team, 0 matches. With two teams, 1 match. With three teams, 2 more matches (making 3). With four teams, 3 more matches (making 6). With five teams, 4 more matches (making 10).
So, following that pattern, 6 teams gives us 15, 7 teams gives us 21, and so on. A little simple addition tells us that 15 teams equals 105 matches.
You want to send a valuable object to a friend securely. You have a box which can be fitted with multiple locks, and you have several locks and their corresponding keys. However, your friend does not have any keys to your locks, and if you send a key in an unlocked box, the key could be copied en route.
How can you and your friend send the object securely?
(Here’s the simplest answer we could come up with. You may very well have come up with alternatives.)
The trick is to remember that you’re not the only one who can put locks on this box.
Put the valuable object into the box, secure it with one of your locks, and send the box to your friend.
Next, have your friend attach one of his own locks and return it. When you receive it again, remove your lock and send it back. Now your friend can unlock his own lock and retrieve the object.
The owner of a winery recently passed away. In his will, he left 21 barrels to his three sons. Seven of them are filled with wine, seven are half full, and seven are empty.
However, the wine and barrels must be split so that each son has the same number of full barrels, the same number of half-full barrels, and the same number of empty barrels.
Note that there are no measuring devices handy. How can the barrels and wine be evenly divided?
For starters, you know your end goal here: You need each set of barrels to be evenly divisible by 3 for everything to work out. And you have 21 barrels, which is divisible by 3. So you just need to move the wine around so make a pattern where each grouping (full, half-full, and empty) is also divisible by 3.
Here’s what you start with:
- 7 full barrels
- 7 half-full barrels
- 7 empty barrels
Pour one of the half-full barrels into another half-full barrel. That gives you:
- 8 full barrels
- 5 half-full barrels
- 8 empty barrels
If you notice, the full and empty barrels increase by one as the half-full barrels decrease by two. (Naturally, the total number of barrels doesn’t change.)
So let’s do it again. Pour one of the half-full barrels into another half-full barrel. That gives you:
- 9 full barrels
- 3 half-full barrels
- 9 empty barrels
And each of those numbers is divisible by 3! Now, each son gets three full barrels, one half-full barrel, and three empty barrels.
How did you do, fellow puzzlers? Did you enjoy Brain Teaser Week? If you did, let us know and we’ll try again with another puzzle genre!
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