Ahh-HAH Puzzles

I love puzzles, but especially what I think of as "ahh-hah" puzzles---puzzles that depend crucially on some logical insight which transforms an impossible question into an obvious answer.

An ha-hah puzzle should be stated, solved, and demonstrated succinctly. Its challenge lies in seeing the ahhh-hah, not in any other heavy lifting: no specialized mathematics, no word-play tricks, no creative real-world thinking, no lengthy derivations, indeed no pencil and paper--just some thinking and a bit of ahhh-hah.

By their nature these puzzles never come in matched sets, they are quirky, they are few and far between. Here are a few that I have collected. Please email me if you enjoy them, or even better, if you have one to add. (Note: in each case I tried to attribute a source to each puzzle, these are not the original sources, rather simply where I first heard it.)

  1. BURNING ROPES

  2. EARTHLY TEMPERATURES

  3. PIRATES LOOT

  4. DROPPING DIMES

  5. THE COMMITTEE WARS

  6. SQUARE MATRIX

  7. A PYRMID SCHEME

  8. EVERYONE PLEASE TAKE A SEAT

  9. INTELLIGENCE DOES COUNT

  10. EVEN CAMEALEONS WANT TO BE DIFFERENT

  11. BILL, FRED, TRUTH, AND LIES

  12. ANTS IN A RING

  13. PLAYING POOL BY THE NUMBERS

  14. TWO CUBED IS EIGHT




































PIRATES LOOT

5 very smart pirates agreed to share 100 diamonds in the following way. Each one of them draws a number (1-5). The pirate gets number 1 proposes a way to distribute diamonds(n1 for #1, n2, for #2,... n1+...+n5=100). Then all pirates(including #1) vote whether to accept the proposal or not. A proposal is passed if half or more than half of the group agree on it. If the proposal is not passed, the one proposed (#1 in first round) will be dumped to ocean as shark food. And then #2 has to propose. It goes one like this until a proposal is accepted. The only goal for pirates is to get more diamond. What's the optimal proposal for pirate #1.

If you find the answer, try 100 pirates and 100 diamonds.

(Source: Mark Brodie)











SQUARE MATRIX

If you have a matrix of positive numbers and all the rows sum to one and all the columns sum to one, prove the matrix is square.











A PYRMID SCHEME

Find the Next line in the follow pyramid 
1
11
21
1211
111221
312211
13112221
1113213211

(Notes: I am embarrassed to place this one in the collection. I just can't get it. I have tried for over an hour on more than one occasion! and I have seen others get it in under a minute. It still qualifies as an ah-hah puzzle I just haven't got the ah-hah yet. Source: Can't remember where I got this from.)











EVERYONE PLEASE TAKE A SEAT

The teacher of a seventh grade class has arranged the students in five rows of five chairs. One day she tell all 25 students to get up and to move to an adjacent unoccupied chair, that is to sit down in a chair directly north, south, east, or west of their current chair.

Can they comply, and if so construct one possible rearrangement.

(Source: From the annual Northern Kentucky University A.H.P. math competition.)











INTELLIGENCE DOES COUNT

A king long ago, wanted to choose the most intelligent among three suitors for his daughter hand. He placed a white dot on each suitor's forehead (where it could not be seen by the suitor). He put them in a room together and told them that the first to infer their dot color was the one to win his daughter's hand. As he left the room he said, "and by the way, at least one of you has a white dot."

About 15 seconds later one of the three declared, "I know my dot's color"

How could he know?

(Source: from memory, I think this is from Scientific American many years ago.)











EARTHLY TEMPERATURES

#1 Prove that there are two diametrically opposed points on the earth's surface that have the same temperature.

NOTE: For this problem assume the earth is a sphere with a continuously varying temperature function mapped over its entire surface.

#2 Prove that there is a pair of diametrically opposed points that simultaneously have the same temperature AND pressure.

(Source: heard at a cocktail party at Lydia Mangu's)











BURNING ROPES

You have an infinite number of ropes that are exactly 12" long, and an infinite supply of matches. Each rope burns in exactly one hour, but not at an even rate. Burn the ropes in a way that allows you to precisely time 45min.

(Source: Given at an IBM T.J. Watson summer student gathering)











EVEN CAMEALEONS WANT TO BE DIFFERENT

As it turns out there are 12 red, 13 white, and 14 blue chameleons on an isolated island. Now unlike your typical variety these creatures shift color *away* from their prior color, and away from the color around them. Thus each time a red and blue lizard meet they both change to the third color (white in this case). If two lizards of the same color just don't notice each other. Obviously if the islands lizards all become the same color through random meeting, they will stay that way. The question is can they all become one color. Provide sequence of meetings or prove that it cannot happen. (Source: Lenny Pitt a prof. and U of Illinois)











BILL, FRED, TRUTH, AND LIES

Bill & Fred are friends, but one always tells the truth and the other always lies. You don't know who is who or who is truthful. Ask one yes/no question to only one of the pair, and determine if Bill is truthful.

(Source: from memory)











ANTS IN A RING

There are 4 ants on the corners of a 10" square. Each ant is facing the ant clockwise from them, and at exactly the same moment they all march directly at the ant in front of them at exactly the same speed. Of course they spiral inward as they do this.

What happens? How many revolutions do they make? How far to they travel? Do they ever collide?

(Source: from memory, seen in multiple places.)











DROPPING DIMES

Suppose we take turns placing 1cm dimes on a 100 x 200 cm rectangular table such that the dime do not fall off the edge and lie flat on the table not touching any other dimes.

Our goal in this game is force your opponent to have no location for their dime. Is there an optimal strategy for this game? If so what is that strategy? does it guarantee a win? and who wins?

(NOTE: These rules can be stated a bit more precisely: Each pair of dimes has some epsilon greater than 0 such that the distances between their centers is 1 cm + epsilon. Source: Lenny Pitt)











PLAYING POOL BY THE NUMBERS

There are 15 balls on a billiard table, bearing the numbers from 1 to 15. Any one of these balls can be selected as the first to go off the table; but thereafter, each subsequent ball must have a number of consecutive (up or down by one) with that of a ball already off the table. How many possible sequences are there for the order in which all 15 ball go off the table?

(Source: I found a scrap of paper on my desk, that someone had photocopied for me. It had this puzzle and the title "Golomb's Puzzle Column" above it.)











TWO CUBED IS EIGHT

A flea is crossing from opposite corners of a two by two by two stack of 1cm sugar cubes. He may only travel on the surfaces of the cubes (including the interior surfaces). What is the length of his shortest path?

(Notes: heard at a cocktail party at Lydia Mangu's)











THE COMMITTEE WARS

A math department of some University has 100 members (wishful thinking, I know :) The department chair decides to form a bunch of committees, as many as possible, but those have to satisfy the following constraints: 1) there is an odd number of people in each committee 2) the intersection between any two committees (the number of people they have in common) must be even.

For example, it is possible to form 100 committees, each having a single person, so that the intersections have 0 people (even number). Another option is to have 100 committees, 99 people in each, 98 people in each intersection. The question is whether it is possible to form MORE than 100 committees.

(Notes: I have not solved this one, but I have only given it a few hours, and those hours were without paper (e.g. driving a car) so I don't know if it rates as an ah-hah puzzle (it might require heavy lifting). This puzzle was forwarded to me with the following message: "This puzzle was mentioned by Vladimir Lifshitz at U of Texas, Austin, http://www.cs.utexas.edu/users/vl/ in the last KR conference - BTW, if you find yourself unable to sleep until 6am because of the puzzle, remember that you are not alone - happened to several people including Vladimir.")

Dan Oblinger