I'm Heron. I coach humans to do math in their heads, I hold the building record for losing at chess to a seven-year-old, and I split those eight balls four against four long before a third pile occurred to me.
Thirty puzzles below. Let's find out how many of them let you keep your first answer.
A balance scale gives three answers, and halving only uses two of them. Left, right, and level: the level reading is free information, and four against four throws it in the bin.
Piles of 3, 3, and 2 use all three. Whichever way the pans go, eight suspects drop to three or two, and one more weighing names the ball.
That is the difference between a puzzle and a riddle. Our math riddles hide the trick in the wording, so reading twice kills them; these thirty hide it in the structure, and reading twice does nothing at all. What gets you through them is the same thing that gets you through a restaurant bill: mental arithmetic that runs without a calculator.
Fun math puzzles to warm up on
Six puzzles, no algebra needed for any of them. The balls up top were puzzle 1, so we pick up at 2.
Keep an honest tally: the scoreboard at the end will ask for it.
Hosting a party?
Riddle 2 of 30
1, 11, 21, 1211, 111221. What comes next in the row?
Hint: Read the line before it out loud, and write down what you hear.
312211
1111221 reads as “three 1s, two 2s, one 1”say the digits, do not add them
2= 312211each line describes the one above it
Every other number row you have seen was built by arithmetic, and this one is built by talking.
Riddle 3 of 30
You have a 3-liter jug and a 5-liter jug. How do you measure exactly 4 liters?
Hint: The useful number is what stays behind, not what you pour out.
4 in the big jug
15 − 3 = 2 left in the big jugfill the 5, pour it into the 3
2empty the 3, move the 2 into itthen refill the 5 to the brim
35 − 1 = 4the small jug takes only 1 more
The jugs cannot show 4, so the leftovers have to, and leftovers only appear after a pour.
Riddle 4 of 30
A ladder hangs off a boat, its lowest rung a foot above the water. What does a 3-foot tide cover?
Hint: The rungs are a foot apart, and nothing is holding the boat down.
1the boat rises with the waterthe ladder rises with the boat
20 rungs coveredthe gap never changes
The tide moves the sea and the boat by the same 3 feet, so the subtraction has nothing to subtract.
Riddle 5 of 30
How many squares of any size can you find on a standard 8×8 chessboard?
Hint: Count the 2×2 squares, then the 3×3 squares, and watch the pattern.
204
✕8 × 8 = 64only the smallest size
11² + 2² + ... + 8²one square size at a time
2= 20464 small, 49 of size two, up to 1 big
Sixty-four answers a smaller question, because a chessboard holds squares in eight different sizes.
Riddle 6 of 30
Place 1 to 9 in a 3×3 grid so every row, column, and diagonal shares one total. What is it?
Hint: Add every number you are given, then look at how many rows they fill.
15
11 + 2 + ... + 9 = 45all nine numbers, once each
245 ÷ 3 rows = 15the rows split the whole pile
3= 15the middle cell has to be 5
The total is fixed before you place a single number, which turns the puzzle into a fitting job.
Riddle 7 of 30
How do you cut a round cake into 8 equal pieces with only 3 straight cuts?
Hint: Two cuts have to live on the top. The third one does not.
Cut it sideways
12 cuts across the top = 4 piecesthe usual cross
21 cut through the middle, flathalves all four at once
34 × 2 = 8two layers of four
Cuts one and two stay in the plane of the table, and the third one leaves it.
The chessboard hides a shape worth keeping. One square size at a time turns the whole board into a short row of numbers to add.
1²+2²+...+5² =
Same rules as above. You've got this.
Math puzzles for kids
These six only need counting, and three of them share one shape. Cuts, posts, and clock strikes all pay out one step away from the number you divide. Three cuts through one pie pay out on their own rule: crossing cuts turn three into seven pieces.
Riddle 8 of 30
A square table has 4 corners. You saw one corner off with a straight cut. How many corners now?
Hint: Draw the cut and count the points where two edges meet.
5
✕4 − 1 = 3a corner cannot just be subtracted
1the cut brings two new cornersone at each edge it crosses
24 − 1 + 2 = 5a five-sided table
One corner goes and two arrive, so a cut that takes something away leaves the table with more.
Riddle 9 of 30
How many cuts do you need to saw a log into 6 pieces?
Hint: Make the first cut and count what you are holding.
5
11 cut = 2 piecesthe first cut buys two
2every later cut adds onepieces run one ahead
36 − 1 = 5five cuts, six pieces
Pieces and cuts are different things, and pieces have a one-step head start.
Riddle 10 of 30
A straight fence is 100 feet long with a post every 10 feet. How many posts does it need?
Hint: Draw the two ends first, then fill in the middle.
11
✕100 ÷ 10 = 10ten spans of fence, not ten posts
1posts = gaps + 1both ends need one
2= 11ten gaps, eleven posts
Dividing the fence counts the empty spans, and the posts outnumber them by the one at the far end.
Riddle 11 of 30
A clock takes 5 seconds to strike 6 o'clock. How long does it take to strike 12?
Hint: The seconds live between the strikes, so count the gaps.
11 seconds
✕5 × 2 = 10 secondsdoubles the strikes, not the gaps
16 strikes = 5 gaps = 1 second each5 seconds split five ways
212 strikes = 11 gaps = 11 sone gap short of twelve
The strikes take no time at all, so the clock is really measuring the silences.
Riddle 12 of 30
How many times does the digit 9 appear in the page numbers from 1 to 100?
Hint: Page 99 is worth more than one tick mark.
20
19, 19, 29, ... 99 = 10the units place
290 to 99 = 10 morethe tens place
310 + 10 = 20page 99 carries two of them
Counting pages and counting digits are two different jobs, and 99 is where they come apart.
Riddle 13 of 30
Three straight cuts down through a pizza, none of them sideways. What is the most pieces?
Hint: Make each new cut cross both of the cuts already there.
7
✕3 cuts = 6 slicesonly if all three meet in the middle
11 + 1 + 2 + 3each cut adds one more than the last
2= 7cross every earlier cut
Staying flat on the table caps you at seven, and the cake in puzzle 7 only got eight by leaving the table.
The log wants 5 cuts for 6 pieces, the fence wants 11 posts for 10 gaps, and the clock wants 11 gaps for 12 strikes. Same shape, three costumes.
We're calling it The Fencepost Gap: the thing you can count and the thing you were asked for sit one step apart. Draw two posts before you divide anything and it dies on the spot.
Middle school math puzzles
Four of these six are percentages, and percentages lie about what they measure. Ten percent off is never the same size as ten percent on.
Ask a share of what before you touch the arithmetic. The speed puzzle and the circle want the same question in their own units.
Riddle 14 of 30
A $100 jacket goes up 10%, then comes down 10% in a sale. What does it cost now?
Hint: The two percents are slices of two different prices.
$99
1100 + 10 = $11010% of a hundred
2110 − 11 = $9910% of a hundred and ten
3= $99a dollar short of where it started
The rise is measured against 100 and the fall against 110, so the two moves never cancel.
Riddle 15 of 30
You drive 60 miles at 30 mph, then 60 miles back at 60 mph. What is your average speed?
Hint: Work out the hours before you touch the speeds.
40 mph
✕(30 + 60) ÷ 2 = 45averages the speeds, not the trip
12 h + 1 h = 3 hthe slow leg costs double
2120 ÷ 3 = 40 mphmiles over hours, as always
You spend twice as long at the slow speed, so it gets twice the vote in the average.
Riddle 16 of 30
Which is bigger: 40% of 60, or 60% of 40?
Hint: Write both of them as a multiplication and look again.
The same
10.4 × 60 = 24the first one
20.6 × 40 = 24the same two numbers
324 = 24a dead heat
A percentage is just multiplication, and multiplication does not care which number wears the sign.
Riddle 17 of 30
A bag holds 4 red and 6 blue marbles. How many red must you add to make red half the bag?
Hint: Every marble you add lands in the total as well.
2
✕half of 10 is 5, so add 1the bag does not stay at 10
1(4 + r) = (10 + r) ÷ 2reds are half of the new total
2r = 26 red, 6 blue
The target moves while you fill the bag, because the marbles you add count on both sides.
Riddle 18 of 30
A square and a circle have the same perimeter. Which one covers more area?
Hint: Give both of them a perimeter of 4 and compare the numbers.
The circle
1square: 1 × 1 = 1perimeter 4, side 1
2circle: π × 0.6366² ≈ 1.27same perimeter 4
31.27 > 1about 27% more area
Corners cost area, and a circle is the one shape that spends none of its perimeter on them.
Riddle 19 of 30
A shirt costs $20 after a 20% discount. What was it before the sale?
Hint: The 20% was cut off a price the tag no longer shows.
$25
✕20 + 20% of 20 = $24adds the percent to the wrong price
10.8 × old = 20$20 is 80% of the old price
2old = $25check it: 25 − 5 = 20
A discount is always measured against the price you can no longer see on the tag.
Percentages are the one place where I still write the base down before I answer. The left-handers below got me twice.
The next six drop the arithmetic almost entirely and ask you to put the moves in the right order.
Math logic puzzles
Small numbers here, and the whole difficulty sits in the order of the moves. A goat rides back across the river, a torch comes back over the bridge, and both trips feel like going backwards.
The habit that clears all six: ask what a move tells you before you count what it costs.
Hosting a party?
Riddle 20 of 30
Wolf eats goat, goat eats cabbage, the boat holds one. How do you ferry all three across?
Hint: One of the three is allowed to make the crossing twice.
Goat first
1goat over, row back emptywolf and cabbage are safe together
2wolf over, bring the goat backthe return trip is a move too
3cabbage over, then the goatseven trips, nothing eaten
Rowing something back looks like losing ground, and it is the only move that solves the puzzle.
Riddle 21 of 30
Four people cross a bridge with one torch. They walk it in 1, 2, 5, and 10 minutes. Fastest total?
Hint: Two people cross at the pace of the slower one, so pair the slow ones up.
17 minutes
11 and 2 cross, 1 comes back2 + 1 = 3 minutes
25 and 10 cross, 2 comes back10 + 2 = 12, running total 15
31 and 2 cross again: 17the 5 hides inside the 10
Sending the fastest walker as everyone's escort costs 19 minutes, because the 5 then gets its own trip.
Riddle 22 of 30
Two doors, one safe. One guard always lies, one always tells the truth. What one question works?
Hint: Make the question travel through both guards before it comes back.
Ask about the other
1“Which door would the other call safe?”ask either one of them
2truth about a lie = a liea lie about the truth is also a lie
3take the other doorboth answers name the wrong one
One guard alone is unreadable, and chaining them makes the lie cancel itself out.
Riddle 23 of 30
How many people do you need in a room to be certain that two of them share a birth month?
Hint: Fill every month with one person and see who is left over.
13
112 months, 12 peoplethey can all miss each other
2person 13 has nowhere newevery month is taken
3= 13one more than the number of months
Certainty costs one person more than the worst case, and the worst case is a perfect spread.
Riddle 24 of 30
Ten bags of coins, real ones 10 g. Every coin in one bag is 1 g light. Find it in one reading.
Hint: Put a different number of coins from every bag on the scale.
Number the bags
11 coin from bag 1, 2 from bag 2, up to 1055 coins in all
2real total = 550 gat 10 g a coin
3550 − reading = the bag3 grams light means bag 3
Numbering the bags turns one reading into ten questions, because each bag now leaves its own dent.
Riddle 25 of 30
You have a 4-minute and a 7-minute hourglass. How do you measure exactly 9 minutes?
Hint: The sand that has already fallen is a timer too.
Flip the 7 twice
1start both, flip the 4 at minute 4the 7 runs on untouched
2at minute 7 flip the 7 as wellthe 4 still has 1 minute in it
3at minute 8 flip the 7 again: 9only 1 minute of sand has fallen
A glass flipped straight back measures the minute it just spent, and that odd minute is the whole trick.
Math puzzles for adults
Five to finish on, and these are the mean ones. A cube with a hidden inner block, a percentage anchored to one stubborn person, and three labels that are all wrong on purpose.
Counting painted cubes means counting edges and corners twice, while counting the clean ones is one multiplication.
Riddle 27 of 30
Two ropes each burn for exactly one hour, but unevenly. How do you measure 45 minutes?
Hint: A rope lit at both ends is done in half its own time, however lumpy it is.
Light three ends
1rope A at both ends, rope B at oneA is gone at minute 30
2light B's second end at 3030 minutes of rope left in it
330 + 15 = 45the rest of B burns in 15
You never learn where the rope burns fast, and lighting both ends makes that ignorance harmless.
Riddle 28 of 30
2, 3, 5, 9, 17. What is the next number in the sequence?
Hint: Look at the jumps between the numbers rather than the numbers.
33
1gaps: 1, 2, 4, 8the jumps are doubling
2next gap = 16so the rule doubles and drops 1
317 + 16 = 33check it: 17 × 2 − 1 = 33
The numbers themselves look like a mess, and their gaps are a clean row of doubles.
Riddle 29 of 30
In a room of 100 people, 99% are left-handed. How many must leave to make it 98%?
Hint: Track the single right-hander, who is not going anywhere.
50
✕99% to 98% is one person1% of a hundred
11 right-hander must become 2%the only person who stays fixed
2room of 50, so 50 leave49 left-handed and 1 right
The percentage that has to move belongs to the one person in the room who never moves.
Riddle 30 of 30
Three boxes: apples, oranges, mixed. Every label is wrong. How many fruits to fix all three?
Hint: Start with the box whose label leaves it the least room to hide.
One
1pull from the box labeled mixedit cannot be mixed
2one fruit names that boxan apple means all apples
3the other two swaptheir labels have only one way left
“Every label is wrong” is a fact worth three inspections, and most solvers read it as scenery.
Speed is what most of them eat. Psychologist Shane Frederick built a famous three-question test out of the same reflex, and at Harvard, MIT, and Princeton most students grabbed the fast, wrong answer.
We first named that reflex The Candy Trap in the Halloween set, and structure puzzles feed on it just as happily as wording ones do. Twelve logic puzzles worked step by step slow the same reflex down to one line at a time.
What to remember
Count the thing you were asked for. The Fencepost Gap catches cuts, posts, strikes, and page digits with the same trick.
Ask a share of what. Percentages only make sense next to the number they were measured against.
Spend a move on information. Three piles beat two, a rope lit at both ends beats a stopwatch, and a goat rowed backwards beats a plan.
Honest tally: how many did you solve without peeking?
0–9
0%
10–19
0%
20–29
0%
All 30
0%
FAQ
Two weighings. Split the balls into piles of 3, 3, and 2, and weigh the two threes against each other. If one side sinks, the heavy ball is in that pile, and weighing any two of its three balls names it. If the pans level out, the heavy ball is one of the two you set aside, and that is your second weighing.
The ones where the arithmetic is easy and the structure is not: the 10×10×10 painted cube, two ropes that burn unevenly, the room where 99% left-handed drops to 98% once fifty people leave, and three boxes whose labels are all wrong. All four are on this page with worked answers.
A riddle hides the trick in the wording, so reading it twice usually kills it. A puzzle hides the trick in the structure: cuts and posts, layers of a cube, the base a percentage is measured against. Reading it twice does nothing, and drawing it out does everything.
Start with the ones that need counting rather than algebra: the square table that gains a corner when you cut one off, how many cuts turn a log into six pieces, how many posts a 100-foot fence needs, how long a clock takes to strike twelve, how often the digit 9 shows up from 1 to 100, and the most slices three cuts can make. Those six are puzzles 8 to 13 here.
They train one narrow habit: pausing on the answer that arrives before you have looked at the shape of the problem. Thirty puzzles is thirty of those pauses, which is more practice than most people get in a month.
