# How Many of These Stupid-Hard Puzzles Can You Solve?

In 2023, I posed 23 challenges as part of the Gizmodo Monday Puzzle series. Some require a bit of math, and others a light familiarity with physics; but most can be solved through logic and creative thinking. For each of the following puzzles, I provide a link to the solution, but I suggest you resist the temptation to look as soon as you feel stumped—the answer may come to you with just a little more contemplation. These challenges are difficult, but they are nearly all solvable without specialized knowledge. Good luck!

You have 10 bags; each bag contains 10 coins. In nine of the bags, every coin weighs exactly 1 gram. However, one of the bags holds counterfeit coins, which weigh 1.1 grams apiece. You don’t know which bag contains the false coins, and the weight difference is too subtle to feel with your hands. Luckily, you have a scale to help. It displays a precise and accurate numeric readout (for example, if you placed a raisin on the scale, it might display: “.58 grams”). Not so luckily, the scale’s battery is dying and after one use it will conk out.

How can you identify the bag with the heavier coins using only a single weighing on the scale? You are allowed to take coins out of their bags, but you only get to read one number from the scale.

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The tic-tac-toe position below was reached in a game between two flawless players, meaning neither of them ever made a move that would allow their opponent to win by force and neither of them missed an opportunity to force a win themselves. What were the last four moves played?

Graphic: Jack Murtagh

You may already know that, with perfect play, tic-tac-toe always ends in a tie where neither player achieves three-in-a-row. So it’s no surprise that a game between two flawless players looks to be headed for stalemate. But how did we get here? To uncover the past, you’ll really need to convince yourself of each move. Producing a plausible-looking sequence of moves will not suffice, because only one such sequence could have occurred under the stipulation of flawless play. Although tic-tac-toe is a small game, the solution requires a sizable dose of deduction.

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Two cubes, each with a single digit between 0 and 9 printed on each face, can be arranged in a holder to display any day of the month as a two-digit number (e.g., 25 on Christmas or 04 on American Independence Day). How do these work? If you were tasked with designing two cubes capable of showing 01, 02, 03, …, 30, 31, how would you assign digits to their faces? This might not seem puzzling, until you set out to try. Remember that cubes only have six faces, so you have 12 total spaces to work with.

If you solve this and crave an additional challenge, how would you represent the three-letter month abbreviations (e.g. ‘jan’, ‘feb’, ‘mar’, and so on) with three cubes, in which each face contains a lowercase letter? Solve the numeric problem first. The same trick will come in handy if you attempt the more difficult letter version.

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To most, Lewis Carroll is best known as the whimsical author of Alice’s Adventures in Wonderland, but did you know that he was also an avid puzzler and published mathematician? Among his many contributions was a book of mathematical puzzles that he called “Pillow Problems.” They are so named because Carroll devised them in bed to distract himself from anxious thoughts while falling asleep.

The challenge: You have an opaque bag containing one marble that has a 50/50 chance of being black or white, but you don’t know which color it is. You take a white marble from your pocket and add it to the bag. Then you shake up the two marbles in the bag, reach in, and pull a random one out. It happens to be white. What are the chances that the other marble in the bag is also white?

Don’t be deceived by the simple setup. This puzzle is famous for defying people’s intuitions. If you struggle to crack it, think it over while falling asleep tonight. It might at least quell your worries.

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Find a one-syllable word that becomes a three-syllable word when one letter is added to the end of it. There are at least three valid answers that are common English words and not proper nouns.Homophones are words that are spelled differently but sound the same. There are few precious examples of four words that are all homophones of each other (like knew, new, gnu, and the Greek letter nu). The puzzle is to find something even rarer: four two-syllable words that are all homophones of each other. Again, no proper nouns.I’m only aware of one satisfying answer to the homophone puzzle (a couple of others only work in some regional accents). Although most readers will have encountered all of the solution words, they may not have realized that they’re spelled differently. Because generating words from thin air is difficult, I’m going to give a hint. If you’d like to attempt the puzzle without a hint, don’t read on.

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Two of the four words in the homophone puzzle are units of measurement. It’s a common misconception that they measure the same thing but, in fact, they’re completely distinct (and spelled differently!)

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Alicia and Bruno are each given a different natural number in secret (1 is the smallest natural number, 2 is the second smallest, and so on). They are then tasked with guessing which of them has the larger number. The following conversation ensues:

Alicia: I don’t know who has the bigger number.

Bruno: I don’t know either.

Alicia: Upon further reflection, I remain ignorant.

Bruno: Alas, I’m still unsure too.

Alicia: Now that you say that, I actually know which of us has the bigger number!

Bruno: Cool! In that case, I know what both of the numbers are.

What numbers were Alicia and Bruno given?

Common knowledge concerns not only what you know, but what you know about what others know about what you know, and so on. It may seem esoteric, but when taken to its logical extreme, common knowledge has bizarre consequences and serves as the basis for many puzzles and even real-world engineering hurdles.

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A gecko is perched on a ceiling corner of a cubic room that measures 10 feet by 10 feet by 10 feet. What is the shortest distance it must travel to reach the diagonally opposite corner of the room? Note that a fly would travel in a straight line along the longest diagonal of the cube, but geckos can only crawl on walls, ceilings, and floors.

I issue a courtesy warning that a dab of geometry crops up in the solution. Don’t let it deter you from discovering the ingenious method. If you remember the Pythagorean theorem, then you know all you’ll need.

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A bat and a ball cost $1.10 in total. The bat costs $1.00 more than the ball. How much does the ball cost?If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets?In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake?These questions are directly quoted from a research paper by Shane Frederick. Do you tend to follow your gut instinct when faced with a new problem? Or are you more likely to take a slow, methodical approach? Everybody uses some mix of these two cognitive modes in their lives, and The Cognitive Reflection Test was designed to measure people’s tendencies toward one or the other. If the answers seemed immediately obvious to you, you may have been relying on your intuitive thinking, when in fact these puzzles require a little more reflection than one might expect. Give them another slower pass and see if any of your answers change.

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You are playing a simple poker game with one deck of cards. Everyone is dealt five cards, and whoever is dealt the best five-card poker hand wins. Lady Luck has promised to deal you a full house of your choice. Which full house should you choose to maximize your chances of winning? Standard rules apply: there are no wild cards, aces can be high or low for straights (10, J, Q, K, A and A, 2, 3, 4, 5 are both valid), but straights cannot “wrap around” (Q, K, A, 2, 3 is not valid). Check here if you need a refresher on poker hand rankings.

I suspect most readers will have a knee-jerk answer for the best full house. Determining why the obvious solution fails will be a key step in solving the puzzle. Or am I just bluffing?

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You have a piping hot cup of coffee that’s too hot to drink. You can either pour a splash of cold milk into it and then let it sit for 10 minutes, or first let it sit for 10 minutes and then add the milk. In which scenario will the coffee end up cooler, or are they equivalent? In both scenarios, assume you pour the same amount of milk and it is the same temperature.You’re in a canoe in the middle of a pond and you brought a rock with you. You pick up the rock, drop it into the water, and watch it sink to the bottom. Does the water level of the pond rise or fall (however imperceptibly) when you do this, or does it stay the same?More knowledge of physics will make these puzzles easier, but you should test your intuitions regardless of your background. Each problem demonstrates a different physical concept, and they have elegant solutions that don’t require calculations. You could even take a proud plunge into empiricism by running the experiments for yourself—please let me know if anybody does this.

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A magic store sells truth-telling machines. The machines each have a yellow light and a blue light that illuminate in response to true-or-false questions, with one light corresponding to “true” and the other to “false.” The store owner has three machines. She tells you that two of them work and tell the truth, but one of them is broken and outputs random answers. She doesn’t know which one is broken. Furthermore, she doesn’t know which light colors correspond to true and false, and the meaning of the lights might differ between machines (i.e., in one machine a yellow light might mean “true,” while in another machine a yellow light might mean “false”). Ask a single true-or-false question to one machine to guarantee that you walk away with a functioning truth-telling machine.

Machines that answer randomly are considerably harder to wrangle than knaves who always lie. Consistent liars mislead in predictable ways! Not knowing what the lights mean and that they can differ across machines poses even more obstacles. While this puzzle is not for the faint of heart, determined mortals can solve it.

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Here are three matchstick puzzles to test your wits.

Graphic: Jack Murtagh

Move two matchsticks so that four one-by-one squares remain. The two matchsticks must be placed somewhere in the diagram and contribute to the final image.

Graphic: Jack Murtagh

This is a fish swimming to the left. Move three matchsticks so that the same fish pattern swims to the right.

Graphic: Jack Murtagh

Above, matchsticks depict numbers like on a digital clock. The equation 3+8=0 is incorrect. Remove only two matchsticks to form a true equation.

Find the solutions at the bottom of this page.

You have five cabinets lined up in a row, and there is a mouse hiding in one of them. Each day, you can open one cabinet to try to find the mouse. If you don’t find it, you close the cabinet, and that night, the mouse moves to one of the cabinets adjacent to it (e.g., if the mouse was hiding in cabinet 2, then it must move to either cabinet 1 or cabinet 3 that night. If the mouse was hiding on the end in cabinet 1, then it must move to cabinet 2 that night, etc.) What sequence of cabinet openings should you use to guarantee that you find the mouse in the fewest number of days, regardless of the mouse’s strategy?

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You are standing somewhere on the surface of the Earth. You walk 1 mile south, then 1 mile west, and then 1 mile north, and you end up back at the same point where you started. Where are you? Assume the Earth is a perfect smooth sphere.

The most natural answer to this question is the North Pole. And it’s correct! Walking a mile south from the North Pole, then a mile west, and a mile north again will land you right back at the top of the planet. According to Musk’s biographer, Ashlee Vance, when engineers would get this correct, Musk would respond, “Where else could you be?” That problem is significantly tougher, and solving it is your task for the week.

The South Pole doesn’t work. Traveling south from the South Pole isn’t even a meaningful concept, since moving south is by definition moving toward the South Pole. It helps to remember that we’re on a big sphere. Think of lines of latitude not actually as horizontal lines but as circles traced around the globe that get progressively wider toward the equator and narrower toward the poles.

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What is the smallest number that contains the letter “a” when spelled out?There is only one number that, when spelled out, has its letters in alphabetical order. What is it? There is also only one number with its letters in reverse alphabetical order. What is it?Imagine we fill a dictionary with the first trillion numbers in alphabetical order. What is the first odd number in the dictionary? This question is tougher than it sounds!Some clarifications:

We’re only considering the counting numbers 1, 2, 3, 4, etc. No fractions, decimals, or negatives.We’re spelling numbers in a canonical way that does not use the word “and.” For example, the number 426 is written “four hundred twenty six.” The number 2,500 must be written “two thousand five hundred” and cannot be written as “twenty five hundred.” 25,000 on the other hand is written “twenty five thousand.” 100 is written “one hundred” and not “a hundred.”Ignore all hyphens and spaces. For example, sixteen comes before six thousand.While some of these might sound like exercises in brute force search, there are strategic ways that you can trim down the search space. Explore the patterns in how we name our numbers and take a stab at whichever prompt you find the most compelling.

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You shuffle a normal deck of cards face down and then begin to flip over cards from the top of the deck, one at a time, placing them face up on a table. At any time (but only once), you can choose to stop, and if the next card is red, then you win. If you never stop, then by default you’re forced to choose the last card (again, you win if it’s red). Is there a strategy that maximizes your chance of winning this game? If so, what is it? If not, why not? You must shuffle the cards thoroughly and are not allowed to cheat in any way (like by marking cards). You may only observe the cards that you flip and choose when to stop.

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Zombies have taken over the world, and you preside over a stronghold with a group of humans trying to build a normal life. You have 1,000 barrels of water to hydrate your people, but your chief scientist comes to you one morning with some disturbing news: there’s been a contamination of the water supply, and one of the 1,000 barrels is now poisonous, but he lost track of which one.

The poisoned water looks and tastes like all of the others, but if anyone drinks even a drop, then they will suddenly fall dead that midnight. You don’t want to risk lives, but since Hollywood has universally agreed that zombies have no moral worth, you are willing to capture and poison them to discover the bad barrel.

Snaring zombies is hard work. What is the fewest number of zombies you’ll need to identify the poisoned barrel by tomorrow?

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There is a hallway with 100 doors labeled 1, 2, 3, etc. up to 100 in order. At first, all of the doors are closed. By toggling a door, I mean changing its position (i.e. opening it if it is closed and closing it if it is open). You walk down the hall and toggle every door (in this case, opening them all). Then a second person walks through and toggles every 2nd door (doors 2, 4, 6, etc.) Then a third person toggles every 3rd door (3, 6, 9, etc.) This continues until the 100th person only toggles the 100th door. Which doors are open at the end?

You could, of course, solve this by brute forcing a simulation of the hallway and merely observing which doors end up open. It will be much more satisfying to think through the problem and discover the reason that a door ends up open or closed.

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1. 100 ants fall onto a meter stick at the same time at random locations. Each ant begins walking toward the left or the right end of the stick at random, at a speed of one meter per minute. Ants continue on their chosen course, but any time two ants collide, they both immediately reverse directions and continue walking the opposite way at the same speed. What is the longest amount of time it could take before all of the ants have walked off the end of the stick?

2. A rectangle is inscribed inside a quarter of a circle that is centered at O. Find the length of the rectangle’s diagonal AC.

Graphic: Jack Murtagh

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Add three straight lines to the diagram to create nine non-overlapping triangles.

Graphic: Jack Murtagh

The triangles may share sides, but shouldn’t share interior space. For example, the left-hand figure below depicts two triangles, whereas the right-hand figure only counts as one triangle, because the larger triangle overlaps with the smaller one.

Graphic: Jack Murtagh

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Which U.S. state is closest to Africa?If you fly due south from Key West, Florida, which South American country will you hit first?I am in an East Coast U.S. state and I call my friend who is in a West Coast U.S. state. It is exactly the same time of day for both of us. How is this possible?The third question requires some specialized knowledge beyond basic geography, but it’s such a cool fact that I had to include it. It’s not a trick question. Try to guess the outline of the solution even without the concrete facts to flesh it out. What types of things would need to be true for such a phone call to take place?

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Albert and Bernard just became friends with Cheryl, and they want to know when her birthday is. Cheryl gives them a list of 10 possible dates.

May 15, May 16, May 19

June 17, June 18

July 14, July 16

August 14, August 15, August 17

Cheryl then tells Albert the month of her birthday and Bernard the day of her birthday separately. Then the following conversation takes place:

Albert: I don’t know when Cheryl’s birthday is, but I know that Bernard also does not know.

Bernard: At first I didn’t know when Cheryl’s birthday is, but I know now.

Albert: Then I also know when Cheryl’s birthday is.

When is Cheryl’s birthday?

You’ll be relieved to hear that the solution does not involve 80-digit numbers. It does, however, require some careful logic.

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You and nine of your coworkers are sitting in a conference room when somebody wonders aloud about the group’s average salary. Everyone is curious, but nobody feels comfortable sharing their own pay with the group. How can you all learn your average salary without anybody learning any more information about another person’s salary (other than what can be deduced from the group average)?

You all have pen and paper and can conceal what you write from others, but you have no other tools at your disposal. You may assume that everybody cooperates with the chosen strategy.

Find the solution here.