Solutions with and without equations, with explicit construction of the pairs
Basic idea
Let F denote the total number of girls and M the total number of boys. In the solutions without equations, we do more than write down the pairs: each time, we show how they arise from the first statement, and then how they change when a brother or a sister becomes the observer.
1. Marta: equal numbers, then double
Marta says: “I have as many sisters as brothers. For each of my brothers, the number of sisters is twice the number of brothers.” How many sons and daughters are there?
Solution with equations
For Marta: F - 1 = M, therefore F = M + 1.
For a brother: F = 2(M - 1).
Substituting: M + 1 = 2M - 2, therefore M = 3 and F = 4.
Answer: 4 girls and 3 boys.
Solution without equations
First statement: Marta does not count herself among her sisters. If she has 1 sister, she must have 1 brother, so the family contains 2 girls and 1 boy: (2,1). If she has 2 sisters and 2 brothers, the pair is (3,2). Continuing: (4,3), (5,4), ...
Second statement: now we look at the family through a brother’s eyes. All the girls remain sisters, whereas he excludes himself from the boys. The pairs therefore become (2,0), (3,1), (4,2), (5,3), ... We seek the case where the number of sisters is twice the number of brothers. This happens with (4,2), because 4 = 2 x 2. Adding the observing brother back gives a total of 3 boys.
Answer: 4 girls and 3 boys.
2. Alberto: the mirror version
Alberto says: “I have as many brothers as sisters. For each of my sisters, the number of brothers is twice the number of sisters.” How many sons and daughters are there?
Solution with equations
For Alberto: M - 1 = F, therefore M = F + 1.
For a sister: M = 2(F - 1).
Substituting: F + 1 = 2F - 2, therefore F = 3 and M = 4.
Answer: 4 boys and 3 girls.
Solution without equations
First statement: Alberto does not count himself among his brothers. If he sees 1 brother and 1 sister, the family contains 2 boys and 1 girl: (2,1), in the order (M,F). Then (3,2), (4,3), (5,4), ...
Second statement: now we look through a sister’s eyes. All the boys remain brothers, whereas she excludes herself from the girls. The pairs become (2,0), (3,1), (4,2), (5,3), ... The pair satisfying the doubling condition is (4,2). Adding the observing sister back gives a total of 3 girls.
Answer: 4 boys and 3 girls.
3. Marta has twice as many sisters as brothers
Marta says: “I have twice as many sisters as brothers.” One of her brothers says: “I have four more sisters than brothers.” How many girls and boys are there?
Solution with equations
For Marta: F - 1 = 2M.
For the brother: F = M + 3.
Therefore 2M + 1 = M + 3: M = 2 and F = 5.
Answer: 5 girls and 2 boys.
Solution without equations
For Marta, the number of sisters is twice the number of brothers. If she has 1 brother, she must have 2 sisters: counting Marta too gives (F,M) = (3,1). If there are 2 brothers, Marta has 4 sisters and the total number of girls is 5: (5,2). Then (7,3), ... A brother, however, does not count himself among the boys. The pairs he sees become (3,0), (5,1), (7,2), ... We seek a difference of 4 between sisters and brothers: with (5,1), we have 5 - 1 = 4. Thus the corresponding family is (5,2).
Answer: 5 girls and 2 boys.
4. Marta has twice as many brothers as sisters
Marta says: “My brothers are twice as numerous as my sisters.” One of her brothers says: “I have as many sisters as brothers.” How many girls and boys are there?
Solution with equations
For Marta: M = 2(F - 1).
For the brother: F = M - 1.
From M = F + 1 and M = 2(F - 1) it follows that F = 3 and M = 4.
Answer: 3 girls and 4 boys.
Solution without equations
If Marta has 1 sister, there must be 2 brothers: counting Marta gives (F,M) = (2,2). If she has 2 sisters, there are 4 brothers: (3,4). If she has 3 sisters, there are 6 brothers: (4,6), ... A brother sees all F girls as sisters, but only M - 1 boys as brothers. The pairs seen become (2,1), (3,3), (4,5), ... The second statement requires equal numbers. This happens with (3,3), which comes from the family (3,4).
Answer: 3 girls and 4 boys.
5. One more sister than brothers
Marta says: “I have one more sister than brothers.” One of her brothers says: “My sisters are twice as numerous as my brothers.” How many girls and boys are there?
Solution with equations
For Marta: F - 1 = M + 1, therefore F = M + 2.
For the brother: F = 2(M - 1).
Therefore M + 2 = 2M - 2: M = 4 and F = 6.
Answer: 6 girls and 4 boys.
Solution without equations
If Marta has 1 brother, she must have 2 sisters; counting Marta too: (F,M) = (3,1). If she has 2 brothers, she must have 3 sisters: (4,2). Then (5,3), (6,4), ... A brother excludes himself from the boys: he therefore sees (3,0), (4,1), (5,2), (6,3), ... We seek the case where the first number is twice the second: 6 is twice 3. The corresponding family is (6,4).
Answer: 6 girls and 4 boys.
6. One more brother than sisters
Alberto says: “I have one more brother than sisters.” One of his sisters says: “My brothers are twice as numerous as my sisters.” How many girls and boys are there?
Solution with equations
For Alberto: M - 1 = F + 1, therefore M = F + 2.
For a sister: M = 2(F - 1).
Therefore F + 2 = 2F - 2: F = 4 and M = 6.
Answer: 6 boys and 4 girls.
Solution without equations
If Alberto has 1 sister, he must have 2 brothers; counting Alberto too: (M,F) = (3,1). If he has 2 sisters, he must have 3 brothers: (4,2). Then (5,3), (6,4), ... A sister excludes herself from the female group: she therefore sees (3,0), (4,1), (5,2), (6,3), ... We need the number of brothers to be twice the number of sisters. 6 is twice 3, so the family is (6,4).
Answer: 6 boys and 4 girls.
7. Three times as many sisters as brothers
Marta says: “I have three times as many sisters as brothers.” One of her brothers says: “I have six more sisters than brothers.” How many girls and boys are there?
Solution with equations
For Marta: F - 1 = 3M.
For the brother: F = M + 5.
Therefore 3M + 1 = M + 5: M = 2 and F = 7.
Answer: 7 girls and 2 boys.
Solution without equations
If Marta has 1 brother, she must have 3 sisters; counting Marta too: (F,M) = (4,1). With 2 brothers, she must have 6 sisters: (7,2). Then (10,3), ... A brother does not count himself: he sees (4,0), (7,1), (10,2), ... We need a difference of 6. With (7,1), the difference is exactly 6, so the original family is (7,2).
Answer: 7 girls and 2 boys.
8. The mirror version of tripling
Alberto says: “I have three times as many brothers as sisters.” One of his sisters says: “I have six more brothers than sisters.” How many girls and boys are there?
Solution with equations
For Alberto: M - 1 = 3F.
For the sister: M = F + 5.
Therefore 3F + 1 = F + 5: F = 2 and M = 7.
Answer: 7 boys and 2 girls.
Solution without equations
If Alberto has 1 sister, he must have 3 brothers; counting Alberto too: (M,F) = (4,1). With 2 sisters: (7,2). Then (10,3), ... A sister does not count herself among the girls: she sees (4,0), (7,1), (10,2), ... We need a difference of 6. With (7,1), the difference is 6: the family is therefore (7,2).
Answer: 7 boys and 2 girls.
9. Three brothers for every sister
Marta says: “I have three brothers for every sister.” One of her brothers says: “I have as many sisters as brothers.” How many girls and boys are there?
Solution with equations
For Marta: M = 3(F - 1).
For the brother: F = M - 1.
From M = F + 1 and M = 3(F - 1) it follows that F = 2 and M = 3.
Answer: 2 girls and 3 boys.
Solution without equations
If Marta has 1 sister, she must have 3 brothers: counting Marta too gives (F,M) = (2,3). If she has 2 sisters, there are 6 brothers: (3,6). Then (4,9), ... A brother excludes himself from the boys: the pairs seen become (2,2), (3,5), (4,8), ... The second statement requires equality. This happens immediately with (2,2), so the family is (2,3).
Answer: 2 girls and 3 boys.
10. Three sisters for every brother
Alberto says: “I have three sisters for every brother.” One of his sisters says: “I have as many brothers as sisters.” How many girls and boys are there?
Solution with equations
For Alberto: F = 3(M - 1).
For a sister: M = F - 1.
From F = M + 1 and F = 3(M - 1) it follows that M = 2 and F = 3.
Answer: 3 girls and 2 boys.
Solution without equations
If Alberto has 1 brother, he must have 3 sisters: counting Alberto too gives (M,F) = (2,3). If he has 2 brothers, there are 6 sisters: (3,6). Then (4,9), ... A sister excludes herself from the girls: she sees (2,2), (3,5), (4,8), ... We need equality between the brothers and sisters seen. It occurs with (2,2), so the family is (2,3), in the order (M,F).
Answer: 3 girls and 2 boys.
11. Two more sisters, then triple
Marta says: “I have two more sisters than brothers.” One of her brothers says: “My sisters are three times as numerous as my brothers.” How many girls and boys are there?
Solution with equations
For Marta: F - 1 = M + 2, therefore F = M + 3.
For the brother: F = 3(M - 1).
Therefore M + 3 = 3M - 3: M = 3 and F = 6.
Answer: 6 girls and 3 boys.
Solution without equations
If Marta has 1 brother, she must have 3 sisters; including Marta, there are 4 girls: (F,M) = (4,1). With 2 brothers: (5,2). With 3 brothers: (6,3), ... A brother excludes himself: he sees (4,0), (5,1), (6,2), ... We need the number of sisters to be three times the number of brothers. 6 is three times 2, so the family is (6,3).
Answer: 6 girls and 3 boys.
12. Two more brothers, then triple
Alberto says: “I have two more brothers than sisters.” One of his sisters says: “My brothers are three times as numerous as my sisters.” How many girls and boys are there?
Solution with equations
For Alberto: M - 1 = F + 2, therefore M = F + 3.
For the sister: M = 3(F - 1).
Therefore F + 3 = 3F - 3: F = 3 and M = 6.
Answer: 6 boys and 3 girls.
Solution without equations
If Alberto has 1 sister, he must have 3 brothers; counting Alberto too, there are 4 boys: (M,F) = (4,1). With 2 sisters: (5,2). With 3 sisters: (6,3), ... A sister excludes herself from the girls: she sees (4,0), (5,1), (6,2), ... We need the number of brothers to be three times the number of sisters. 6 is three times 2: the family is (6,3).
Answer: 6 boys and 3 girls.
13. Double plus one
Marta says: “The number of my sisters is twice the number of brothers, plus one.” One of her brothers says: “I have five more sisters than brothers.” How many girls and boys are there?
Solution with equations
For Marta: F - 1 = 2M + 1, therefore F = 2M + 2.
For the brother: F = M + 4.
Therefore 2M + 2 = M + 4: M = 2 and F = 6.
Answer: 6 girls and 2 boys.
Solution without equations
If Marta has 1 brother, the number of sisters is 2 x 1 + 1 = 3; counting Marta gives (F,M) = (4,1). With 2 brothers, there are 5 sisters and 6 girls in total: (6,2). With 3 brothers: (8,3), ... A brother excludes himself from the boys: he sees (4,0), (6,1), (8,2), ... We need a difference of 5. With (6,1), the difference is exactly 5, so the family is (6,2).
Answer: 6 girls and 2 boys.
14. Mirror version: double plus one
Alberto says: “The number of my brothers is twice the number of sisters, plus one.” One of his sisters says: “I have five more brothers than sisters.” How many girls and boys are there?
Solution with equations
For Alberto: M - 1 = 2F + 1, therefore M = 2F + 2.
For the sister: M = F + 4.
Therefore 2F + 2 = F + 4: F = 2 and M = 6.
Answer: 6 boys and 2 girls.
Solution without equations
If Alberto has 1 sister, the number of brothers is 2 x 1 + 1 = 3; counting Alberto gives (M,F) = (4,1). With 2 sisters: (6,2). With 3 sisters: (8,3), ... A sister excludes herself from the girls: she sees (4,0), (6,1), (8,2), ... We need a difference of 5. With (6,1), the difference is 5, so the family is (6,2).
Answer: 6 boys and 2 girls.
15. Twice as many brothers, then one fewer
Marta says: “My brothers are twice as numerous as my sisters.” One of her brothers says: “I have one fewer sister than brothers.” How many girls and boys are there?
Solution with equations
For Marta: M = 2(F - 1).
For the brother: F = (M - 1) - 1 = M - 2.
Therefore M = F + 2 and F + 2 = 2F - 2.
It follows that F = 4 and M = 6.
Answer: 4 girls and 6 boys.
Solution without equations
First statement: we start with the number of Marta’s sisters. If Marta has 1 sister, there are 2 x 1 = 2 brothers; counting Marta too, there are 2 girls. We therefore obtain (F,M) = (2,2). If Marta has 2 sisters, there are 4 brothers and 3 girls in total: (3,4). If she has 3 sisters, there are 6 brothers and 4 girls in total: (4,6). Continuing: (5,8), ...
Second statement: now we look through a brother’s eyes. All the girls remain sisters, whereas he does not count himself among the brothers. The pairs therefore become (2,1), (3,3), (4,5), (5,7), ... We seek the case where the brother has one fewer sister than brothers. With (4,5), he sees 4 sisters and 5 brothers: the condition is satisfied. The corresponding family pair is therefore (4,6).
Answer: 4 girls and 6 boys.