Easy math puzzles
Poll
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51 members have voted
February 26th, 2026 at 4:25:45 PM
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Let's say that you start with 15 ml of milk and 250 ml of coffee. You move one teaspoon (5 ml) of milk into the coffee mug.
The ratio of coffee to total volume becomes 250/255 which is about 98%. Then, you take one teaspoon (5ml) of the 98% coffee + 2% milk mixture and dump it into the milk container. This teaspoon is 98% coffee and 2% milk, so .98*5ml coffee = 4.9 ml and 0.1 ml milk.
The milk container had 10 ml of milk after a teaspoon of it was removed. After adding back 5 ml, its volume is again 15 ml, and most of that, but not all, that is returned (98%) is coffee.
The amount of coffee in the milk cup is 4.901961 ml. The amount of milk in the coffee cup is 5 ml.
The ratio of coffee to total volume becomes 250/255 which is about 98%. Then, you take one teaspoon (5ml) of the 98% coffee + 2% milk mixture and dump it into the milk container. This teaspoon is 98% coffee and 2% milk, so .98*5ml coffee = 4.9 ml and 0.1 ml milk.
The milk container had 10 ml of milk after a teaspoon of it was removed. After adding back 5 ml, its volume is again 15 ml, and most of that, but not all, that is returned (98%) is coffee.
The amount of coffee in the milk cup is 4.901961 ml. The amount of milk in the coffee cup is 5 ml.
February 26th, 2026 at 7:22:44 PM
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Quote: KevinAALet's say that you start with 15 ml of milk and 250 ml of coffee. You move one teaspoon (5 ml) of milk into the coffee mug.
The ratio of coffee to total volume becomes 250/255 which is about 98%. Then, you take one teaspoon (5ml) of the 98% coffee + 2% milk mixture and dump it into the milk container. This teaspoon is 98% coffee and 2% milk, so .98*5ml coffee = 4.9 ml and 0.1 ml milk.
The milk container had 10 ml of milk after a teaspoon of it was removed. After adding back 5 ml, its volume is again 15 ml, and most of that, but not all, that is returned (98%) is coffee.
The amount of coffee in the milk cup is 4.901961 ml. The amount of milk in the coffee cup is 5 ml.
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I'm not sure where your mistake is, but here are the amounts and ratios after each movement.
After first movement
| Cup | Coffee | Milk | Total | Ratio Coffee | Ratio Milk |
|---|---|---|---|---|---|
| Coffee | 250 | 5 | 255 | 0.980392 | 0.019608 |
| Milk | 0 | 10 | 10 | 0.000000 | 1.000000 |
| Cup | Coffee | Milk | Total | Ratio Coffee | Ratio Milk |
|---|---|---|---|---|---|
| Coffee | 245.0980392 | 4.901960784 | 250 | 0.980392 | 0.019608 |
| Milk | 4.901960784 | 10.09803922 | 15 | 0.326797 | 0.673203 |
Note the milk in the coffee cup = coffee in the milk cup = 4.901960784.
We at least agree there is 4.901960784 ml of coffee in the milk cup. That had to displace the same amount of milk that used to be there. Where else could it have gone but the coffee cup?
"My life is spent in one long effort to escape from the commonplace of existence. These little problems help me to do so." -- Sherlock Holmes
February 26th, 2026 at 8:54:27 PM
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OK, I get it now. That was a good one!
February 27th, 2026 at 7:17:34 AM
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You flip a fair coin 50 times. What is the probability of never seeing three or more heads in a row? A numeric answer to six significant digits will suffice.
"My life is spent in one long effort to escape from the commonplace of existence. These little problems help me to do so." -- Sherlock Holmes
February 27th, 2026 at 10:21:16 AM
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Exact answer should be the 52nd tribonacci number divided by 2^50. There is a closed form formula for tribonaccis but it’s much longer than the Fibonacci formula.
Very easy to Markov the answer though I consider that brute forcing it
Very easy to Markov the answer though I consider that brute forcing it
It’s all about making that GTA
February 27th, 2026 at 10:32:28 AM
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This was about the 500th puzzle on this thread that wasn’t “easy”Quote: WizardI've been working on this on my own. I found this video, by one of my favorite YouTube channels, to be very useful in finding the nth term in the Fibonacci sequence. From there, I can see how to adapt the logic for the dice question.
Direct: https://www.youtube.com/watch?v=ITSbuT9ojOw
By the way, this was hardly an Easy math puzzle.
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It’s all about making that GTA
February 27th, 2026 at 10:38:59 AM
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Quote: Ace2Exact answer should be the 52nd tribonacci number divided by 2^50. There is a closed form formula for tribonaccis but it’s much longer than the Fibonacci formula.
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It does not help that determining a closed form for a recursive sequence that depends on the three previous terms requires finding the roots to a cubic equation.
February 27th, 2026 at 11:28:41 AM
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I'm still looking for a simple Markov Chain answer, in part to verify my own answer.
"My life is spent in one long effort to escape from the commonplace of existence. These little problems help me to do so." -- Sherlock Holmes
February 27th, 2026 at 1:13:45 PM
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Here are some riddles for you.
- What has a face and hands but no arms or legs?
- If I have a bee in my hand, what is in my eye?
- What's harder for you to catch the faster you run?
- I appear twice in the morning. I appear twice in the evening. But I only appear once at night. What am I?
- What has 13 hearts but no other organs?
- What has three feet but can't walk?
- What has a tail but no body?
- What is so delicate that if you say its name, it breaks?
- What is always on its way but never arrives?
- Five friends are together in a room. Charlie is knitting. Don is cooking. Ace is playing chess. Gordon is reading a book. What is the fifth friend doing?
"My life is spent in one long effort to escape from the commonplace of existence. These little problems help me to do so." -- Sherlock Holmes
February 27th, 2026 at 1:48:21 PM
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What has a face and hands but no arms or legs?
An analog clock
If I have a bee in my hand, what is in my eye?
What's harder for you to catch the faster you run?
Your breath
I appear twice in the morning. I appear twice in the evening. But I only appear once at night. What am I?
The letter N
What has 13 hearts but no other organs?
A deck of cards
What has three feet but can't walk?
A yardstick
What has a tail but no body?
A coin (usually)
What is so delicate that if you say its name, it breaks?
Silence
What is always on its way but never arrives?
Tomorrow
Five friends are together in a room. Charlie is knitting. Don is cooking. Ace is playing chess. Gordon is reading a book. What is the fifth friend doing?
Watching one of those videos on Facebook where you click on the link and it brings up a page loaded with ads but it never does reveal the answer to what was on the original video
February 27th, 2026 at 2:55:05 PM
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Very simple Markov chain with only three states (last 2, 1, or 0 flips were heads). The sum of the three states after fifty iterations is the answer…gives you the probability that there were never more than two consecutive headsQuote: WizardI'm still looking for a simple Markov Chain answer, in part to verify my own answer.
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It’s all about making that GTA
February 27th, 2026 at 3:35:02 PM
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I haven't completed the spreadsheet but here is a possible approach. It considers fifty empty slots which are to be filled with "H" or "T", in such a way that there aren't any "HHH"..
Consider three kinds of lines
(a) lines which end "HH"
(b) lines which end "H"
(c) lines which do not have a "H" in last position
Then, initially, fill the remainder of the line with tails.
Now consider two kinds of additions
(i) "HT" (one head)
(ii) "HHT" (two heads)
Note for the each series of "Heads" to finish, there must be room for a "Tail" after it/them. As we're only interested in one or two Heads, then these are the only two types of additions allowed. Since there is a possiblity, at the end of a line, for none, one or two Heads not followed by a Tail, then looks at these types of lines - which now have 48,49,50 "empty" slots.
For each type of line how many of (i) or (ii) can one add
"HH" will have 48 slots, "H" will have 49, "none" will have 50
Another idea is to use similar logic for (say) a row of ten, identifying how many start or end with any Heads (i.e. whether they could be put next to each other). (I might look at this a bit more later!)
Consider three kinds of lines
(a) lines which end "HH"
(b) lines which end "H"
(c) lines which do not have a "H" in last position
Then, initially, fill the remainder of the line with tails.
Now consider two kinds of additions
(i) "HT" (one head)
(ii) "HHT" (two heads)
Note for the each series of "Heads" to finish, there must be room for a "Tail" after it/them. As we're only interested in one or two Heads, then these are the only two types of additions allowed. Since there is a possiblity, at the end of a line, for none, one or two Heads not followed by a Tail, then looks at these types of lines - which now have 48,49,50 "empty" slots.
For each type of line how many of (i) or (ii) can one add
"HH" will have 48 slots, "H" will have 49, "none" will have 50
| "HT" | "HHT" | check | 50 slots | 49 slots | 48 slots |
| 0 | 0 | 0 | 1 | 1 | 1 |
| 0 | 1 | 3 | 48 | 47 | 46 |
Another idea is to use similar logic for (say) a row of ten, identifying how many start or end with any Heads (i.e. whether they could be put next to each other). (I might look at this a bit more later!)
February 27th, 2026 at 4:32:17 PM
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Using brute force one can work out how many of the 1024 ways to toss coins results in the block of ten results starting with a tail or consecutive heads.
| (a) | T - T | 149 |
| (b) | T - H | 81 |
| (c) | T - HH | 44 |
| (d) | H - T | 81 |
| (e) | H - H | 44 |
| (f) | H - HH | 24 |
| (g) | HH - T | 44 |
| (h) | HH - H | 24 |
| (i) | HH - HH | 13 |
The next table looks at how many ways to end the first ten. Thus 1T says it ends with a tail, which can be a d or g. Similarly for H or HH.
| a/d/g | 1T | 274 |
| b/e/h | 1H | 149 |
| c/f/i | 1HH | 81 |
The next table looks at how you can move from 1T,1H,1HH to various other states. For instance to get from 1HH to 2HH you can only have a T-HH, as the others would create three Heads.
| a/d/g | 1T > 2T | 274 | 274 | 75076 |
| b/e/h | 1T > 2H | 274 | 149 | 40826 |
| c/f/i | 1T > 2HH | 274 | 81 | 22194 |
| a/d | 1H > 2T | 149 | 230 | 34270 |
| b/e | 1H > 2H | 149 | 125 | 18625 |
| c/f | 1H > 2HH | 149 | 68 | 10132 |
| a | 1HH > 2T | 81 | 149 | 12069 |
| b | 1HH > 2H | 81 | 81 | 6561 |
| c | 1HH > 2HH | 81 | 44 | 3564 |
You then just repeat this logic working with how far you've got and observing that 2T = 1T>2T+1H>2T+1HH>2T etc.
| a/d/g | 2T > 3T | 121415 | 274 | 33267710 |
| b/e/h | T >2H | 121415 | 149 | 18090835 |
| c/f/i | T > HH | 121415 | 81 | 9834615 |
| a/d | H > T | 66012 | 230 | 15182760 |
| b/e | H >2H | 66012 | 125 | 8251500 |
| c/f | H > HH | 66012 | 68 | 4488816 |
| a | HH >2T | 35890 | 149 | 5347610 |
| b | HH > H | 35890 | 81 | 2907090 |
| c | HH > HH | 35890 | 44 | 1579160 |
| a/d/g | 3T > 4T | 53798080 | 274 | 14740673920 |
| b/e/h | T >2H | 53798080 | 149 | 8015913920 |
| c/f/i | T > HH | 53798080 | 81 | 4357644480 |
| a/d | H > T | 29249425 | 230 | 6727367750 |
| b/e | H >2H | 29249425 | 125 | 3656178125 |
| c/f | H > HH | 29249425 | 68 | 1988960900 |
| a | HH >2T | 15902591 | 149 | 2369486059 |
| b | HH > H | 15902591 | 81 | 1288109871 |
| c | HH > HH | 15902591 | 44 | 699714004 |
| a/d/g | 4T > 5T | 23837527729 | 274 | 6531482597746 |
| b/e/h | T >2H | 23837527729 | 149 | 3551791631621 |
| c/f/i | T > HH | 23837527729 | 81 | 1930839746049 |
| a/d | H > T | 12960201916 | 230 | 2980846440680 |
| b/e | H >2H | 12960201916 | 125 | 1620025239500 |
| c/f | H > HH | 12960201916 | 68 | 881293730288 |
| a | HH >2T | 7046319384 | 149 | 1049901588216 |
| b | HH > H | 7046319384 | 81 | 570751870104 |
| c | HH > HH | 7046319384 | 44 | 310038052896 |
| TOTAL | 19426970897100 |
This gives the total ways. Just divide by 250 to get the chances.
February 27th, 2026 at 5:09:22 PM
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Just keep count of the various states as the rolls roll in. The lost is two times (as the roll could be a head or tail) the running total plus HH (as they threw a Head).
| Roll | …T | …H | …HH | Lost |
| 1 | 1 | 1 | 0 | 0 |
| 2 | 2 | 1 | 1 | 0 |
| 3 | 4 | 2 | 1 | 1 |
| 4 | 7 | 4 | 2 | 3 |
| 5 | 13 | 7 | 4 | 8 |
| 6 | 24 | 13 | 7 | 20 |
| 7 | 44 | 24 | 13 | 47 |
| 8 | 81 | 44 | 24 | 107 |
| 9 | 149 | 81 | 44 | 238 |
| 10 | 274 | 149 | 81 | 520 |
| 11 | 504 | 274 | 149 | 1121 |
| 12 | 927 | 504 | 274 | 2391 |
| 13 | 1705 | 927 | 504 | 5056 |
| 14 | 3136 | 1705 | 927 | 10616 |
| 15 | 5768 | 3136 | 1705 | 22159 |
| 16 | 10609 | 5768 | 3136 | 46023 |
| 17 | 19513 | 10609 | 5768 | 95182 |
| 18 | 35890 | 19513 | 10609 | 196132 |
| 19 | 66012 | 35890 | 19513 | 402873 |
| 20 | 121415 | 66012 | 35890 | 825259 |
| 21 | 223317 | 121415 | 66012 | 1686408 |
| 22 | 410744 | 223317 | 121415 | 3438828 |
| 23 | 755476 | 410744 | 223317 | 6999071 |
| 24 | 1389537 | 755476 | 410744 | 14221459 |
| 25 | 2555757 | 1389537 | 755476 | 28853662 |
| 26 | 4700770 | 2555757 | 1389537 | 58462800 |
| 27 | 8646064 | 4700770 | 2555757 | 118315137 |
| 28 | 15902591 | 8646064 | 4700770 | 239186031 |
| 29 | 29249425 | 15902591 | 8646064 | 483072832 |
| 30 | 53798080 | 29249425 | 15902591 | 974791728 |
| 31 | 98950096 | 53798080 | 29249425 | 1965486047 |
| 32 | 181997601 | 98950096 | 53798080 | 3960221519 |
| 33 | 334745777 | 181997601 | 98950096 | 7974241118 |
| 34 | 615693474 | 334745777 | 181997601 | 16047432332 |
| 35 | 1132436852 | 615693474 | 334745777 | 32276862265 |
| 36 | 2082876103 | 1132436852 | 615693474 | 64888470307 |
| 37 | 3831006429 | 2082876103 | 1132436852 | 130392634088 |
| 38 | 7046319384 | 3831006429 | 2082876103 | 261917705028 |
| 39 | 12960201916 | 7046319384 | 3831006429 | 525918286159 |
| 40 | 23837527729 | 12960201916 | 7046319384 | 1055667578747 |
| 41 | 43844049029 | 23837527729 | 12960201916 | 2118381476878 |
| 42 | 80641778674 | 43844049029 | 23837527729 | 4249723155672 |
| 43 | 148323355432 | 80641778674 | 43844049029 | 8523283839073 |
| 44 | 272809183135 | 148323355432 | 80641778674 | 17090411727175 |
| 45 | 501774317241 | 272809183135 | 148323355432 | 34261465233024 |
| 46 | 922906855808 | 501774317241 | 272809183135 | 68671253821480 |
| 47 | 1697490356184 | 922906855808 | 501774317241 | 137615316826095 |
| 48 | 3122171529233 | 1697490356184 | 922906855808 | 275732407969431 |
| 49 | 5742568741225 | 3122171529233 | 1697490356184 | 552387722794670 |
| 50 | 10562230626642 | 5742568741225 | 3122171529233 | 1106472935945520 |
| Winners | 19426970897100 |
February 27th, 2026 at 5:17:40 PM
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Quote: charliepatrickJust keep count of the various states as the rolls roll in. The lost is two times (as the roll could be a head or tail) the running total plus HH (as they threw a Head).
Roll …T …H …HH Lost 1 1 1 0 0 2 2 1 1 0 3 4 2 1 1 4 7 4 2 3 5 13 7 4 8 6 24 13 7 20 7 44 24 13 47 8 81 44 24 107 9 149 81 44 238 10 274 149 81 520 11 504 274 149 1121 12 927 504 274 2391 13 1705 927 504 5056 14 3136 1705 927 10616 15 5768 3136 1705 22159 16 10609 5768 3136 46023 17 19513 10609 5768 95182 18 35890 19513 10609 196132 19 66012 35890 19513 402873 20 121415 66012 35890 825259 21 223317 121415 66012 1686408 22 410744 223317 121415 3438828 23 755476 410744 223317 6999071 24 1389537 755476 410744 14221459 25 2555757 1389537 755476 28853662 26 4700770 2555757 1389537 58462800 27 8646064 4700770 2555757 118315137 28 15902591 8646064 4700770 239186031 29 29249425 15902591 8646064 483072832 30 53798080 29249425 15902591 974791728 31 98950096 53798080 29249425 1965486047 32 181997601 98950096 53798080 3960221519 33 334745777 181997601 98950096 7974241118 34 615693474 334745777 181997601 16047432332 35 1132436852 615693474 334745777 32276862265 36 2082876103 1132436852 615693474 64888470307 37 3831006429 2082876103 1132436852 130392634088 38 7046319384 3831006429 2082876103 261917705028 39 12960201916 7046319384 3831006429 525918286159 40 23837527729 12960201916 7046319384 1055667578747 41 43844049029 23837527729 12960201916 2118381476878 42 80641778674 43844049029 23837527729 4249723155672 43 148323355432 80641778674 43844049029 8523283839073 44 272809183135 148323355432 80641778674 17090411727175 45 501774317241 272809183135 148323355432 34261465233024 46 922906855808 501774317241 272809183135 68671253821480 47 1697490356184 922906855808 501774317241 137615316826095 48 3122171529233 1697490356184 922906855808 275732407969431 49 5742568741225 3122171529233 1697490356184 552387722794670 50 10562230626642 5742568741225 3122171529233 1106472935945520 Winners 19426970897100
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I agree!
The answer can also be found by doing a modified Fibonacci series, starting with 2-4-7 and going back the last three terms.
"My life is spent in one long effort to escape from the commonplace of existence. These little problems help me to do so." -- Sherlock Holmes
February 27th, 2026 at 5:18:23 PM
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Quote: ThatDonGuy
What has a face and hands but no arms or legs?
An analog clock
If I have a bee in my hand, what is in my eye?
What's harder for you to catch the faster you run?
Your breath
I appear twice in the morning. I appear twice in the evening. But I only appear once at night. What am I?
The letter N
What has 13 hearts but no other organs?
A deck of cards
What has three feet but can't walk?
A yardstick
What has a tail but no body?
A coin (usually)
What is so delicate that if you say its name, it breaks?
Silence
What is always on its way but never arrives?
Tomorrow
Five friends are together in a room. Charlie is knitting. Don is cooking. Ace is playing chess. Gordon is reading a book. What is the fifth friend doing?
Watching one of those videos on Facebook where you click on the link and it brings up a page loaded with ads but it never does reveal the answer to what was on the original video
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Agreed. Still looking for 2 and 10, although points given for humor on #10.
"My life is spent in one long effort to escape from the commonplace of existence. These little problems help me to do so." -- Sherlock Holmes
February 27th, 2026 at 5:53:25 PM
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That’s actually called the tribonacci series. To calculate the probability of no consecutive quads you’d use the tetranacci series, and so onQuote: WizardQuote: charliepatrickJust keep count of the various states as the rolls roll in. The lost is two times (as the roll could be a head or tail) the running total plus HH (as they threw a Head).
Roll …T …H …HH Lost 1 1 1 0 0 2 2 1 1 0 3 4 2 1 1 4 7 4 2 3 5 13 7 4 8 6 24 13 7 20 7 44 24 13 47 8 81 44 24 107 9 149 81 44 238 10 274 149 81 520 11 504 274 149 1121 12 927 504 274 2391 13 1705 927 504 5056 14 3136 1705 927 10616 15 5768 3136 1705 22159 16 10609 5768 3136 46023 17 19513 10609 5768 95182 18 35890 19513 10609 196132 19 66012 35890 19513 402873 20 121415 66012 35890 825259 21 223317 121415 66012 1686408 22 410744 223317 121415 3438828 23 755476 410744 223317 6999071 24 1389537 755476 410744 14221459 25 2555757 1389537 755476 28853662 26 4700770 2555757 1389537 58462800 27 8646064 4700770 2555757 118315137 28 15902591 8646064 4700770 239186031 29 29249425 15902591 8646064 483072832 30 53798080 29249425 15902591 974791728 31 98950096 53798080 29249425 1965486047 32 181997601 98950096 53798080 3960221519 33 334745777 181997601 98950096 7974241118 34 615693474 334745777 181997601 16047432332 35 1132436852 615693474 334745777 32276862265 36 2082876103 1132436852 615693474 64888470307 37 3831006429 2082876103 1132436852 130392634088 38 7046319384 3831006429 2082876103 261917705028 39 12960201916 7046319384 3831006429 525918286159 40 23837527729 12960201916 7046319384 1055667578747 41 43844049029 23837527729 12960201916 2118381476878 42 80641778674 43844049029 23837527729 4249723155672 43 148323355432 80641778674 43844049029 8523283839073 44 272809183135 148323355432 80641778674 17090411727175 45 501774317241 272809183135 148323355432 34261465233024 46 922906855808 501774317241 272809183135 68671253821480 47 1697490356184 922906855808 501774317241 137615316826095 48 3122171529233 1697490356184 922906855808 275732407969431 49 5742568741225 3122171529233 1697490356184 552387722794670 50 10562230626642 5742568741225 3122171529233 1106472935945520 Winners 19426970897100
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I agree!
The answer can also be found by doing a modified Fibonacci series, starting with 2-4-7 and going back the last three terms.
link to original post
It’s all about making that GTA
