# The children of the sultan

An old challenge from an old Portuguese book I have:

Abdul Al-Kwaritz-Amuhmahd was a Arabic sultan who became very famous by the size of his harem and the number of children he had fathered. More so because many of his children were twins, triplets or quadruplets.
One of the legends about this, tells that all the children of the sultan were twins, except 39 of them.
On a remote oasis, in the middle of the desert, another saying states that all of them were triplets, except 39.
The sultan descendants defend, however, that all were quadruplets except 39.

Ajihad Ahmed-Humahli Hijá, an investigator, says that all the three states about the children of the sultan are true. If he is right, hom many children did the sultan have fathered?

For the purists: the sultan had, at least one "normal" child (not a twin, nor a triplet, nor a quadruplet) and, at least, one pair of twins.

• You have mentioned, "the sultan had, at least one "normal" child (not a twin, nor a triplet, nor a quadruplet) and, at least, one pair of twins." I cannot understand the purpose of this line. The answer that I just posted would be the same, with or without this condition. Request you or someone to let me know the purpose of the above line. Thanks . Commented Aug 7, 2022 at 9:02
• @HemantAgarwal Otherwise "39 normal children, 0 twins, 0 triplets and 0 quadruplets" would be a valid solution. Although I suppose that would still clash with the first paragraph which says that "many" of his children were twins, triplets and quadruplets.
– Jafe
Commented Aug 7, 2022 at 11:24
• @Jafe, thanks for the clearing!
– Pspl
Commented Aug 13, 2022 at 11:35

12 quadruplets, 12 twins, and 12 triplets would be sufficient to have an equal number of children in quadruplets, triplets, and twins. Since this is 12 per group, the other two groups will be within the 39. So, 39-(12*2) = 39-24 = 15 children remaining.

Thus, Abdul Al-Kwaritz-Amuhmahd's children were (cheekily):

6 pairs of twins,
4 groups of triplets,
2 groups of sextuplets, and
3 only children

for a total of

51 children

• That was fast :) You nailed it. Just waiting to accept your answer :) Although you don't have to include sextuplets on your answer, eh eh!
– Pspl
Commented Nov 11, 2020 at 17:19

Let's assume that the sultan has $$2x$$ twins, $$3y$$ triplets and $$4z$$ quadruplets (since the numbers must be divisible by 2, 3 and 4 respectively). He has also $$t$$ "normal" children. Now, we have the Diophantine equation system: $$t+2x+3y=39$$, $$t+2x+4z=39$$, $$t+3y+4z=39$$ ($$t\geqslant1$$, $$x\geqslant1$$, $$y\geqslant0$$, $$z\geqslant0$$).
Now we have $$2x+3y=2x+4z=3y+4z=39-t$$, so $$2x=3y=4z=\frac{39-t}2$$. That means that $$\frac{39-t}2$$ is divisible by 12, but it must be between $$0$$ and $$19=\frac{39-1}2$$ (since $$t\geqslant1$$). So, $$2x=3y=4z=12$$ ($$x=6$$, $$y=4$$, $$z=3$$) and $$t=39-24=15$$. Total number of children is $$12+12+12+15=51$$ (6 pair of twins, 4 triplets, 3 quadruplets and 15 "single" children).

• +1 - This is a formalization of the intuitive process I went through to get to my answer.
– Avi
Commented Nov 11, 2020 at 17:25

Let number of single births be W
twin births be X
triplet births be. Y

Therefore, total number of children = W+2X+3Y+4Z

Let us assume that no other births were there i.e no 5 children were born together, etc.

W+3Y+4Z = 39 (Equation 1)
W+2X+4Z = 39 (Equation 2)
W+2X+3Y = 39 (Equation 3)

Equation 2 - Equation 3 gives us
4Z-3Y = 0

This means that number of children who are born as quadruplets are equal to number of children who are born as triplets.

Now, let us look at equation 1 again.
W+3Y+4Z = 39
We know that the number of quadruplets has to be equal to the number of triplets. Therefore, the only possibility of quadruplets and triplets that will satisfy this equation is 3Y= 4Z = 12

W = 39-12-12 = 15

This solves the entire question.
15 children born as as a single child.
12 children born as twins.
12 children born as triplets.

The sultan, therefore, has 51 ( 15+12+12+12) children.

Note : I cannot understand the purpose of this line, "The sultan had, at least one "normal" child (not a twin, nor a triplet, nor a quadruplet) and, at least, one pair of twins."

Even if the above line wasn't mentioned, the answer would still have been 15, 12, 12 and 12. Request someone to explain to me, the purpose of the above line.

• Since it's pre-faced with "for the purists", I guess it's for pedantry purpose? Like having 39 single-birth children and 0 twins, triplets, quadruplets? Commented Aug 7, 2022 at 10:46
• Third time right. The first was handwavy ("12" magically appeared from nowhere); the second was a wall of algebra; and this one was both well explained and nicely formatted. Commented Aug 7, 2022 at 13:41
• Minor nit-pick: the 15 don't have to be singles (e.g. 3 sets of quints). Commented Aug 7, 2022 at 13:43
• Nit-pick on nit-pick: 2 quints, yes, 3 quints no, since there is at least one single child. Commented Aug 7, 2022 at 15:26