Discussion in 'Physics & Math' started by ProCop, Jan 8, 2005.

1. ProCopValued Senior Member

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1,258

1, 3, 5, 9, 15, 31, 61, 125, 251, 503, 1015, 2035, 4081, 8177, 16367, 32747, ???

This row of numbers was posted some time ago at www.fok.nl. TS didn't know the sollution, nor did anyone else come with any. The thread went dead - it was concluded that there have had to be a mistake somewhere eg. that the numbers were badly copied. (There is wel same traceable pattern but it is not constant/regulair)

Possibly someone here might solve it. We'll see...

3. FortunaRegistered Senior Member

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Hmm, interesting series Procop. My first observation is that they are remarkably close to the powers of 2.

2^n for the first element(1)
2^n +1 for the next 3 elements(3,5,9)
2^n - 1 for the next 2 elements(15,31)
2^n -3 for the next 2 (61 and 125)
2^n - 5 for 251
2^n - 9 for 503 and 1015
etc

My guess is that the form is something like 2^n + f(n) , where I dont yet know what f(n) is.

5. ProCopValued Senior Member

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1,258
A function could be a way to tackle it,

but I came accross an another (possible) pattern:

1
3
5
<b>9 </b>
15
31
61
<b>125</b>

the bold numbers are sums of all previous numbers, but this regurality is not forwarded further in the string...

7. ProCopValued Senior Member

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1,258
A riddle

This riddle was sent to a prestigious puzzle competition by a girl - and it won the competition:

Katja was celebrating her birthday and two days <b> later</b> her <b>older </b> twin sister Romy (they were one egg twins) celibrated her birthday. How come?

8. invert_nexusZe do caixaoValued Senior Member

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9,686
Leap year.
Katja was born on the 28th of February before midnight.
Romy was born on the 1st of March after midnight.
Easy.

Reminds me of a comedy version of the Pirates of Penzance (sp?) called the Pirate Movie from the 80's. The young hero is freed from his vow of loyalty to the pirate king because of his 18th birthday, but later the Pirate King presses him back in service because he was born on the 29th of February and therefore he's only 4 years old.

Edit:
Wait. That doesn't work, does it?
Hmm.
Does it have something to do with a time machine and a contraceptive?

Could always be due to relativity, but I don't think that's the answer you're looking for.

Edit again:
International date line maybe? Would that spread it out to two days later? It could depending on the exact time of each celebration.

Last edited: Jan 24, 2005
9. ProCopValued Senior Member

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1,258
invert_nexus very good!!

(the ship on which they were born crossed the timezone-line during the interval between the two births.... (so that the older girl was born in March while the younger one (the second born) was born on 28th February))

10. ProCopValued Senior Member

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1,258
A king had three sons. 100 of his offspring had each 2 children the rest had none. Wat was the total of the king' s offspring?

11. QuarkHeadRemedial Math StudentValued Senior Member

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10<sup>2</sup> = 100

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14. DilbertRegistered Senior Member

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361
hmm, the king had three sons. But then it says that 100 of the (kings???) offspring had each 2 children and the rest had none.

Well, it says how many sons he has, not how many children. And the 100 offspring integer has not reached its maximum, it just says that 100 of his offspring and not how many the offspring was limited to.

Either the answer is 3, or it cannot be answered. Because it is impossible to know if those 3 were a part of those 100, and there is no way of knowing if there are more than 100 offsprings. And frankly i do not know if "his" means the king or not.

15. ProCopValued Senior Member

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1,258
OK more specifics:

The king has only the three sons (none other children)
The king's offspring include all of his children and their children and so on and exactly 100 of these ofspring (the total number of kings ofspring must include also all ofspring which possibly had no children) have 2 children (each individual offspring of the 100 has 2 children one offspring can have 2 children or none children only (thus not one child)) .

The total number of the kings offspring is sollution to the puzzle.

16. QuarkHeadRemedial Math StudentValued Senior Member

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Well - in that case we all excuse ourselves! To some of us
a) only 3 sons does not mean the same as 3 sons only
b)offspring refers only to the direct issue of your loins (if I may put that way) - not to grandchildren as well

17. MacMRegistered Senior Member

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I would think that another factor could be that the problem does not either specifiy or preclude any daughters of the king?

To be absolutely sure the problem is not a symantical one can you state that the king has three sons, no daughters or morphadites?

Last edited: Jan 25, 2005
18. DilbertRegistered Senior Member

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MacM, you do not have to exaggerate it. I think that all agree that it was difficult to interpret, but that does not means that we have to have it explained to us in absurdum.

19. ProCopValued Senior Member

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"Offspring" in the frame of this riddle means all children grandchildren grandgrandchildren as far as necesaary, sex of the children is not importand in the context of the riddle either, it is mainly logic, little maths and can be figured out without a pen and paper, though you can write it down if you wish...

20. MacMRegistered Senior Member

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10,104

Maybe morpadite was a bit over the top. But my point was to seek verification that it is not symantical but mathematical in its nature.

21. shmoeRegistred UserRegistered Senior Member

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524
In white:

Suppose only 1 member of each generation after the king has children. So one of his sons produces the others don't, one of his 2 grandchildren produces, the other doesn't, and so on. Each generation (except the last) has one child that produces, so it will take 101 generations after the king to have 100 productive offspring. Each generation has 2 people in it, except the first which has 3), so 101*2+1=203 offspring.

Is this a unique answer? Could a different arrangement satisfy the problem and give a different number of offspring? A little graph theory will clear this up. Consider the kings family tree (starting at him) as a tree in the graph theory sense, the people are represented as verticies and the parent-child relationships represented by the edges. Let e be the number of edges and v the number of vertices. In any tree e+1=v. We also know 2*e=sum of degrees of vertices, where the degree of a vertex is the number of edges it touches. The king has degree 3 (3 sons), every one of his offspring that produces children has degree 3 (2 children and one parent), there are 100 vertices like this. Every non-productive offspring has degree 1 (it's parent), let's say there are x of these. So we have e+1=v=1+100+x and 2*e=3+3*100+x. Solve for x to get x=103. So he had 100 offspring who produced, 103 who didn't and the # of offspring is unique-this had no reliance on the shape of the tree.

This is assuming no incest. If you allow the cousin lovin then the family tree is no longer a tree and the above does not apply.

Alternate method:

The king celebrates 2*100 births of grandchildren and beyond, plus his own 3 sons. So he has 200+3=203 offspring. Again, this assumes no incest.

Either method is still under my definition of 'little math' and pen&paper free.

22. phoenix2634Registered Senior Member

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um, is the answer 300? So, what if 3 of the kings offspring were sons. He had 100 offspring who had 2 kids each and the others (200 of em) had none.

23. DilbertRegistered Senior Member

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Phoenix, 303? i do not think you included the original children.

But i do not know, i think it's a poorly formulated riddle.