One way to approach this question is to think about how many of each digit you would need. Let's start with 0 and move up.
0: We need two 0-slates to make numbers like 500 or 600. The worst case scenario is that the four numbers we get are 100, 200, 300, and 400, or any combination with eight 0s altogether. Thus, we need eight 0s.
1: Suppose one of our choices is 111. The other three choices cannot be 111, but they can all include two 1s. So the number of 1s we need is 3 + 2 + 2 + 2 = 9.
2: Similarly, we need nine of these.
3: Again, nine
4: Again, nine.
5: Nine!
6: Nine!
7: Nine!
8: First, note that we never have to make 888. So we don't need to consider the worst case scenario that we considered for 1. However, we can still have a situation where all four numbers contain two 8s, in which case the number of 8s we need is 2 * 4 = 8.
9: This is the tricky part. Remember that the guy said we can make a 9 with a 6. The worst case scenario involving 9s and 6s would be when all four numbers have three digits and the digits only include 9 and 6: for example, {666, 669, 696, 699}. In this case, we need 4 * 3 = 12 of either 6 or 9. Note that we already have nine 6s. So we only need 3 more.
Add them up. We need eight 0s and eight 8s. We need nine of each of 1, 2, 3, 4, 5, 6, and 7. Finally, we need three 9s.
8 * 2 + 9 * 7 + 3 = 16 + 63 + 3 = 82.
Answer: 82
Monday, February 17, 2014
Sunday, February 16, 2014
Dante's Infernal Puzzle Collection -- 19. Grana
Suppose he sold x of units of the poor grains and y units of the fair grains.
His cost was 32x + 40y.
He sold x + y units of grains at the price of 43, so his revenue was 43x + 43y.
(32x + 40y) * (1.25) = 43x + 43y
40x + 50y = 43x + 43y
3x = 7y
x = (7/3)y
Suppose x + y = 100.
(7/3)y + y = 10/3y = 100
y = 30
x = 100 - 30 = 70
Thus, 70% of the grains he sold were poor, and 30% were fair.
Answer: 70% poor, 30% fair
His cost was 32x + 40y.
He sold x + y units of grains at the price of 43, so his revenue was 43x + 43y.
(32x + 40y) * (1.25) = 43x + 43y
40x + 50y = 43x + 43y
3x = 7y
x = (7/3)y
Suppose x + y = 100.
(7/3)y + y = 10/3y = 100
y = 30
x = 100 - 30 = 70
Thus, 70% of the grains he sold were poor, and 30% were fair.
Answer: 70% poor, 30% fair
Dante's Infernal Puzzle Collection -- 18. Domanda
The goal is to find out which gesture means yes and which means no. Ask a question whose answer must be yes -- for example, "Is this Hell?"
Wednesday, February 5, 2014
Dante's Infernal Puzzle Collection -- 16. Frutta
We will use the following notation:
A: weight of one apple
L: weight of one plum
P: weight of one peach
According to the information given to us, we know:
1) 3A + 1P = 10L
2) 1P = 6L + 1A
And we need to solve for x in the following equation.
1P = xL
Recall that 1P = 6L + 1A. so xL = 6L + 1A.
From equation 1) above, 3A + xL = 10L, and thus 3A = (10-x)L, which means 1A = [(10-x)/3]L.
Plugging this back into the above equation (xL = 6L + 1A), we get xL = 6L + [(10-x)/3]L.
Multiply both sides by 3 --> 3xL = 18L + 10L - xL.
Collect terms --> 4xL = 28L
x = 7.
Answer: 7 plums.
A: weight of one apple
L: weight of one plum
P: weight of one peach
According to the information given to us, we know:
1) 3A + 1P = 10L
2) 1P = 6L + 1A
And we need to solve for x in the following equation.
1P = xL
Recall that 1P = 6L + 1A. so xL = 6L + 1A.
From equation 1) above, 3A + xL = 10L, and thus 3A = (10-x)L, which means 1A = [(10-x)/3]L.
Plugging this back into the above equation (xL = 6L + 1A), we get xL = 6L + [(10-x)/3]L.
Multiply both sides by 3 --> 3xL = 18L + 10L - xL.
Collect terms --> 4xL = 28L
x = 7.
Answer: 7 plums.
Dante's Infernal Puzzle Collection -- 15. Demochares
Denote Demochares's age at death by x.
From the information provided in the question, we know:
(1/4)x + (1/5)x + (1/3)x + 13 = x
(1/4 + 1/5 + 1/3)x + 13 = x
[(15 + 12 + 20)/60]x + 13 = x
(47/60)x + 13 = x
13 = (13/60)x
x = 60
Answer: 60
From the information provided in the question, we know:
(1/4)x + (1/5)x + (1/3)x + 13 = x
(1/4 + 1/5 + 1/3)x + 13 = x
[(15 + 12 + 20)/60]x + 13 = x
(47/60)x + 13 = x
13 = (13/60)x
x = 60
Answer: 60
Dante's Infernal Puzzle Collection -- 14. Pianto
(The book's answer is wrong again. Let me explain why later.)
The answer is that we need more information.
Suppose there were originally 200 women and 100 men. Then the number of women in the pit would be 200 * .42 = 84, whereas the number of men would be 100 * .28 = 28. The male-to-female ratio is 28:84, or 1:3.
Now suppose there were initially 100 women and 200 men. The number of women in the pit is 100 * .42 = 42, and the number of men in the pit is 200 * .28 = 56. The male-to-female ratio is 56:42, which is equivalent to 4:3.
The point is, the answer completely depends on the initial number of men and women. Without this information, we cannot provide an answer to the question.
The book's solution gives an extremely simplistic answer by simply taking the ratio of 28 to 42. This is incorrect.
The answer is that we need more information.
Suppose there were originally 200 women and 100 men. Then the number of women in the pit would be 200 * .42 = 84, whereas the number of men would be 100 * .28 = 28. The male-to-female ratio is 28:84, or 1:3.
Now suppose there were initially 100 women and 200 men. The number of women in the pit is 100 * .42 = 42, and the number of men in the pit is 200 * .28 = 56. The male-to-female ratio is 56:42, which is equivalent to 4:3.
The point is, the answer completely depends on the initial number of men and women. Without this information, we cannot provide an answer to the question.
The book's solution gives an extremely simplistic answer by simply taking the ratio of 28 to 42. This is incorrect.
Dante's Infernal Puzzle Collection -- 13. Cruna D'Ago
The spirit initially had 200 florins. Let x = 200.
Each year, the spirit's wealth increased by .5 of the previous amount.
At the end of the first year, he had 1.5x.
At the end of the second year, he had 1.5 * 1.5x.
At the end of the third year, he had 1.5 * 1.5 * 1.5x... and so on.
So eighteen years later, the spirit had (1.5^18)x = 1477.8919 * 200 = 295,578.376
Answer: 295,578.376 florins
Each year, the spirit's wealth increased by .5 of the previous amount.
At the end of the first year, he had 1.5x.
At the end of the second year, he had 1.5 * 1.5x.
At the end of the third year, he had 1.5 * 1.5 * 1.5x... and so on.
So eighteen years later, the spirit had (1.5^18)x = 1477.8919 * 200 = 295,578.376
Answer: 295,578.376 florins
Saturday, February 1, 2014
Dante's Infernal Puzzle Collection -- 12. Spericolato
In the amount of time it took the horse rider to vanish around the turn, the man walked 52 paces.
Also note that it took the man 52 + 312 = 364 paces to cover the same distance.
d = the distance from the point of their encounter to the turn
t = the time it took for the horse rider to vanish around the turn = the time it took the man to walk 52 paces
Note that it took the man (364/52) * t to reach the circle.
Since [speed = distance / time], the speed of the horse rider is d/t, whereas the speed of the man is d/(364/52)t.
Since the man claims that his speed is 2 leagues per hour, we can set up the problem as follows:
d/(364/52)t : 2 = d/t : x, where x is the horse rider's speed.
2 * (d/t) = xd/(364/52)t
2d = xd/(364/52)
2d = (52xd)/364
2 = 52x/364
52x = 728
x = 14
Thus, the horse rider's speed is 14 leagues per hour.
Answer: 14 leagues/hr
Dante's Infernal Puzzle Collection -- 11. Filosofi
Notice that only one of them is telling the truth.
1) Suppose Euclid is the truth-teller.
Then it must be that Euclid was telling the truth when he said he's not the greatest one.
We also know that Plato is not the greatest one, because Socrates, who is a liar according to our assumption, says that he is.
Note that Plato asserts that Socrates is wrong. Since our assumption indicates that Plato's claim is a lie, it must be that Socrates is right. This, however, isn't the case, because it violates our assumption that Socrates is not telling the truth.
This contradiction leads to the conclusion that Euclid is not the truth-teller.
2) Suppose Socrates is the truth-teller.
Since Euclid is lying, we know that his claim is wrong and thus Euclid is the greatest.
Remember that we're assuming that Socrates is telling the truth. So his claim that Plato is the greatest must be true.
However, it cannot be that Plato and Euclid are both the greatest.
The assumption that Socrates is the truth-teller also leads to a contradiction.
3) Finally, suppose Plato is the truth-teller.
W know that this has to be the case, because the other two assumptions didn't work.
Since Euclid is lying, his claim that he's not the greatest is wrong -- in other words, Euclid is the greatest.
Since Socrates is also lying, we know that Plato is not the greatest.
Plato is telling the truth, and his claim that Socrates is wrong is correct.
We find no contradiction here, and Euclid is the greatest.
Answer: Euclid
Dante's Infernal Puzzle Collection -- 10. Giorni
One day after the day before yesterday is yesterday.
Two days after yesterday is tomorrow.
Answer: tomorrow
Two days after yesterday is tomorrow.
Answer: tomorrow
Dante's Infernal Puzzle Collection -- 9. Charon
(Fisrt, note that the book's solution is extremely unconvincing. The outcome would depend on the ratios that we use. See the note below to see why.)
The three sisters are names Anna, Carla, and Maria. Denote Anna's speed as As, Carla's speed as Cs, and Maria's speed as Ms. Similarly, denote Anna's thrift as At, Carla's as Ct, and Maria's as Mt. Lastly, denote Anna's warmth as Aw, Carla's as Cw, and Maria's as Mw.
Anna claims that she made 5 shirts while Carla made 2. Then Carla says that while Anna made 3 shirts, Maria made 4. Normalizing As = 1, we get Cs = 2/5 and Ms = 4/3.
Normalize At = 1. Next we are informed that Mt = 1/5 and Ct = 3 * (1/5) / 5 = 3/25.
Normalizing Aw = 1, we also find out that Cw = 1/3 and Mw = (1/3) * (1/4) = 1/12.
We are not given any directions regarding how to weigh these scores, but I assumed that for each individual, you take the inverse of her thrift score and add her warmth and speed scores to it. For example, Maria's score would be Ms + (1/Mt) + Mw = 4/3 + 1/(1/5) + 1/12 = 4/3 + 5 + 1/12 = 6.xxx.
Similarly, Carla's score would be Cs + (1/Ct) + Cw = 2/5 + 1/(3/25) + 1/3 = 136/15 = 9.xxx.
Anna's score is As + (1/At) + Aw = 1 + 1 + 1 = 3, which is actually the lowest.
So, according to my scoring guideline...
Answer: Not true
Note:
As mentioned above, the book offers not only an erroneous solution, but an extremely simplistic one at best. First, it states that the ratio of warmth is (Maria : Anna : Carla) = 3.75 : 60 : 15. This is wrong. Anna's score should be 3 times that of Carla's, but 60 is clearly not 15*3.
More importantly, the winner depends on how you take the ratios. Instead of 15 : 3 : 25 for lightness, suppose we multiply each score by 100. Then the ratio would be 1500 : 300 : 2500. Now calculate each person's total score. Maria's score is 20 + 1500 + 3.75 = 1523.75. Anna's score is 15 + 300 + 60 = 375, which clearly indicates that she performed worse than Maria.
The book's solution seems to have resulted from a misguided understanding of ratios. We cannot arbitrarily standardize ratios, add up the individual values, and use the outcome as a standard for comparison.
The three sisters are names Anna, Carla, and Maria. Denote Anna's speed as As, Carla's speed as Cs, and Maria's speed as Ms. Similarly, denote Anna's thrift as At, Carla's as Ct, and Maria's as Mt. Lastly, denote Anna's warmth as Aw, Carla's as Cw, and Maria's as Mw.
Anna claims that she made 5 shirts while Carla made 2. Then Carla says that while Anna made 3 shirts, Maria made 4. Normalizing As = 1, we get Cs = 2/5 and Ms = 4/3.
Normalize At = 1. Next we are informed that Mt = 1/5 and Ct = 3 * (1/5) / 5 = 3/25.
Normalizing Aw = 1, we also find out that Cw = 1/3 and Mw = (1/3) * (1/4) = 1/12.
We are not given any directions regarding how to weigh these scores, but I assumed that for each individual, you take the inverse of her thrift score and add her warmth and speed scores to it. For example, Maria's score would be Ms + (1/Mt) + Mw = 4/3 + 1/(1/5) + 1/12 = 4/3 + 5 + 1/12 = 6.xxx.
Similarly, Carla's score would be Cs + (1/Ct) + Cw = 2/5 + 1/(3/25) + 1/3 = 136/15 = 9.xxx.
Anna's score is As + (1/At) + Aw = 1 + 1 + 1 = 3, which is actually the lowest.
So, according to my scoring guideline...
Answer: Not true
Note:
As mentioned above, the book offers not only an erroneous solution, but an extremely simplistic one at best. First, it states that the ratio of warmth is (Maria : Anna : Carla) = 3.75 : 60 : 15. This is wrong. Anna's score should be 3 times that of Carla's, but 60 is clearly not 15*3.
More importantly, the winner depends on how you take the ratios. Instead of 15 : 3 : 25 for lightness, suppose we multiply each score by 100. Then the ratio would be 1500 : 300 : 2500. Now calculate each person's total score. Maria's score is 20 + 1500 + 3.75 = 1523.75. Anna's score is 15 + 300 + 60 = 375, which clearly indicates that she performed worse than Maria.
The book's solution seems to have resulted from a misguided understanding of ratios. We cannot arbitrarily standardize ratios, add up the individual values, and use the outcome as a standard for comparison.
Dante's Infernal Puzzle Collection -- 8. Aggruppamento
Ten packs, one thousand people in total.
We can rephrase this question as "What are the ten natural numbers with which, by summing one or more of them, we can make any integer in [1, 1000]?"
Let's approach this sequentially -- we will start with 1 and see if it must be included in our group of 10 natural numbers. Then we will think about 2, 3,..., 1000 in the same way.
First, we know that 1 has to be one of those ten numbers -- otherwise, we wouldn't be able to make 1.
We also know that 2 needs to be there as well. If we skip 2, there's no way we can make 2.
3 doesn't have to be included, because we can make 3 by adding 1 and 2.
So far, we only have 1 and 2 on our list, which means the highest number we can form is 1 + 2 = 3. This means that we need to include 4 in our list. Thus the list becomes {1, 2, 4}.
We already know that 1, 2, and 3 can be formed with {1, 2}. Adding these to 4, we know that we can create any number in [1, 7] with {1, 2, 4}. But since we can't make 8 yet, we need to add 8 to the list.
Now it's just induction. We know that any number in [1, 7] can be formed with just {1, 2, 4}. This implies that with {1, 2, 4, 8}, we can make any number in [1, 7+8], or [1, 15]. So we add 16 to the list, which now becomes {1, 2, 4, 8, 16}. Continuing this way, the final list is {1, 2, 4, 8, 16, 32, 64, 128, 256, 512}. However, recall that the sum of all the 10 numbers should be 1000, which means we should have 1000 - 1 - 2 - 4 - 8 - 16 - 32 - 64 - 128 - 256 = 489 instead of 512.
Notice that Virgil says that four of those numbers are needed to form 300. Those four numbers would be 4, 8, 32, and 256.
Notice that Virgil says that four of those numbers are needed to form 300. Those four numbers would be 4, 8, 32, and 256.
Answer: 1, 2, 4, 8, 16, 32, 64, 128, 256, 489
Dante's Infernal Puzzle Collection -- 7. Predatore
Google tells me that one league is approximately 18228 feet.
The hound runs at a speed of 12 leagues an hour, or 1/5 league a minute.
In 5 minutes, the hound runs 1 league.
Now let us examine the progress of the fowl. Since its initial speed is 20 leagues/hr, in the first quarter-minute it runs 20/(4*60) = 20/240 of a league. In the second quarter-minute, its speed becomes 19 leagues/hr, which is equivalent to 19/(4*60) = 19/240 league per quarter-minute.
Thus in 5 minutes, the fowl runs:
20/240 + 19/240 + ... + 1/240 = (20 + ... + 1)/240 = (21*10)/240 = 7/8 of a league.
Recall that the hound, on the other hand, runs 1 league = 18228 feet in those 5 minutes, which is significantly greater than 7/8 of a league + 10 feet = 15949.5 + 10 feet = 15959.5 feet.
Hence we conclude that the hound catches the fowl.
Answer: Yes
The hound runs at a speed of 12 leagues an hour, or 1/5 league a minute.
In 5 minutes, the hound runs 1 league.
Now let us examine the progress of the fowl. Since its initial speed is 20 leagues/hr, in the first quarter-minute it runs 20/(4*60) = 20/240 of a league. In the second quarter-minute, its speed becomes 19 leagues/hr, which is equivalent to 19/(4*60) = 19/240 league per quarter-minute.
Thus in 5 minutes, the fowl runs:
20/240 + 19/240 + ... + 1/240 = (20 + ... + 1)/240 = (21*10)/240 = 7/8 of a league.
Recall that the hound, on the other hand, runs 1 league = 18228 feet in those 5 minutes, which is significantly greater than 7/8 of a league + 10 feet = 15949.5 + 10 feet = 15959.5 feet.
Hence we conclude that the hound catches the fowl.
Answer: Yes
Dante's Infernal Puzzle Collection -- 6. Fedelta
I had to look at the answer for this one. The answer is simply that the stranger was the couple's newborn child.
Dante's Infernal Puzzle Collection -- 5. Retaggio
Suppose you give the older one x and the younger one y.
Since the initial amount was 100, it must be that x + y = 100.
The question states as follows:
(1/4)x - (1/3)y = 11
Multiply both sides by 12.
3x - 4y = 132.
Now just do the math.
x + y = 100
3x - 4y = 132.
Solving results in 7y = 168.
Thus y = 24, and x = 100 - y = 100 - 24 = 76.
Answer: 76 to the elder, 24 to the younger
Since the initial amount was 100, it must be that x + y = 100.
The question states as follows:
(1/4)x - (1/3)y = 11
Multiply both sides by 12.
3x - 4y = 132.
Now just do the math.
x + y = 100
3x - 4y = 132.
Solving results in 7y = 168.
Thus y = 24, and x = 100 - y = 100 - 24 = 76.
Answer: 76 to the elder, 24 to the younger
Dante's Infernal Puzzle Collection -- 4. Segare
Again, simple algebra.
Gathering one heap takes 1/5 of an hour.
Packing one heap takes 1/8 of an hour.
Suppose we gather x heaps. Since we need to fill an hour, it must be the case that:
1/5x + 1/8x = 1
Multiply both sides by 40.
8x + 5x = 40
13x = 40
x = 40/13
Answer: 40/13
Gathering one heap takes 1/5 of an hour.
Packing one heap takes 1/8 of an hour.
Suppose we gather x heaps. Since we need to fill an hour, it must be the case that:
1/5x + 1/8x = 1
Multiply both sides by 40.
8x + 5x = 40
13x = 40
x = 40/13
Answer: 40/13
Dante's Infernal Puzzle Collection -- 3. Virgil
For simplicity, let's fervently hope that the initial explanation from one of the men was true...
Notice that Dante asks the second one if he is a truth-teller. There are only two possibilities, which are as follows:
1) Suppose the first one is a truth-teller and the second one is a liar. Since the first one always tells truth, it must be true that the second one would indeed say "yes" to Dante's question. Since the second one is a liar, this is indeed the case -- he would in fact say yes, despite the fact that the true answer is no. We find no contradiction here.
2) Now suppose the first one is a liar and the second one is a truth-teller. This means that the first one's assertion that the second one would say "yes" is a lie -- in other words, the second one would actually say "no." This, however, leads to a contradiction. If the second one is a truth-teller, his answer should be yes. Hence we rule out the possibility that the first one lies and the second one tells truth.
Answer: Trust the first one.
Notice that Dante asks the second one if he is a truth-teller. There are only two possibilities, which are as follows:
1) Suppose the first one is a truth-teller and the second one is a liar. Since the first one always tells truth, it must be true that the second one would indeed say "yes" to Dante's question. Since the second one is a liar, this is indeed the case -- he would in fact say yes, despite the fact that the true answer is no. We find no contradiction here.
2) Now suppose the first one is a liar and the second one is a truth-teller. This means that the first one's assertion that the second one would say "yes" is a lie -- in other words, the second one would actually say "no." This, however, leads to a contradiction. If the second one is a truth-teller, his answer should be yes. Hence we rule out the possibility that the first one lies and the second one tells truth.
Answer: Trust the first one.
Dante's Infernal Puzzle Collection -- 2. Sacchetto
Name the three sacks A, B, and C, and their correct contents a, b, and c. Suppose you open A. Since all labels were misplaced, there are only two possibilities: 1) you see b, or 2) you see c. Without loss of generality, suppose A contains b.
This means that B must contain c. Otherwise we end up violating the rule -- if B contains a, then C would be left with c, violating the assumption that all labels have been misplaced. If B contains b, this again violates the same assumption.
Thus, we are left with a for C.One investigation is sufficient to reveal the contents of the sacks.
Answer: one.
This means that B must contain c. Otherwise we end up violating the rule -- if B contains a, then C would be left with c, violating the assumption that all labels have been misplaced. If B contains b, this again violates the same assumption.
Thus, we are left with a for C.One investigation is sufficient to reveal the contents of the sacks.
Answer: one.
Dante's Infernal Puzzle Collection -- 1. Fontana
Elementary algebra. Essentially, the question asks: "If 1/4 of 20 becomes 4, what does 10/3 become?
1/4 of 20 is 5.
5 became 4 --> the initial value (5) has been reduced to 4/5 of itself.
Thus we multiply 10/3 by 4/5 --> 10/3 * 4/5 = 8/3.
Answer: 8/3
1/4 of 20 is 5.
5 became 4 --> the initial value (5) has been reduced to 4/5 of itself.
Thus we multiply 10/3 by 4/5 --> 10/3 * 4/5 = 8/3.
Answer: 8/3
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