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magicjohn2278
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AT LAST ONE TO BEAT THOSE MATHS EXPERTS!


My friend showed me a strange dice with the numbers :

1, 8, 17, 18, 27 and 27

..on it. When I asked him about his dice and the significance of the numbers he said that the numbers were very special, and it wasn't a dice or a die, but a cube.

Do the numbers have any special significance?
magicjohn2278
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It would help if I learned to type! The numbers should be:

1, 8, 17, 18, 27 and 28

(It still works the original way but this is better!)
magicjohn2278
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... called in to see my friend again yesterday to find out more about his cube. Didn't get very far, he seemed absorbed in a problem involving four frogs!

He muttered something about "adding the digits of the cube.." So I did.... thirty-six? I don't see any significance in this, except that it is the square of the number of sides on his cube.... but this problem has nothing to do with squares?...
magicjohn2278
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...no takers? ....All the clues are in the question.......
stanalger
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Add the digits of the cube of each number...and you get the original number
again.

1^3=1
8^3=512 and 5 + 1 + 2 = 8
17^3=4913 and 4 + 1 + 9 + 3=17
18^3=5832 and 5+ 8 + 3 + 2 = 18
27^3=19683 and 1 + 9 + 6 + 8 + 3 = 27
28^3=21952 and 2 + 1 + 9 + 5 + 2 = 28
stanalger
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What about 26?
Why wasn't it invited to the party?

26^3=17576 and 1+7+5+7+6=26
magicjohn2278
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See, the clues ARE in the question (somewhere!)

But well done Stanalger! I was going to post a note in the answer saying that these were the ONLY six numbers that do this. The puzzle is based on one that I saw in a cheap puzzle book! (A pretty basic "What do the following have in common?")- This is the second piece of mis-information that I have got from there!
magicjohn2278
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.... anyway, who ever heard of a cube with SEVEN sides!?
TomasB
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I didn't understand the part with

"2 + 1 + 9 + 5 + 2 = 28"

Someone care to explain?

/Tomas
magicjohn2278
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Oops! .... Moral of the story is "Don't rely on cheap puzzle books!"

The originl question from the book is:

These numbers have a strange property, can you see what it is?

1 8 17 18 27 28

(Hint - think cubes!)

The answer was given that these were the ONLY six numbers where the sum of the digits of the square equaled the original number.

... So I was somewhat surprised when Stanalger found another! The numbers are a misprint, and not the first that I have found! I'll check the answers in future!

(Actually there is another that I am thinking of posting, but I'm not convinced that the solution given is correct - yet!)
stanalger
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Quote:
On 2006-03-16 16:00, TomasB wrote:
I didn't understand the part with

"2 + 1 + 9 + 5 + 2 = 28"

Someone care to explain?

/Tomas


Tomas,

The explanation is simple: I was careless.

28= 2 + 19 + 5 + 2 ....but that's not what we're talking about here, is it?
Since we're not allowed to "clump" digits, 28 should not be in the list.

The complete list of natural numbers whose cubes have a digit-sum equal to the
original integer is 1, 8, 17, 18, 26, 27. (Unless you consider 0 to be
a natural number.)

In bases other than base-ten, the list would of course be different.

Looks like John's puzzle book erroneously replaced the 26 with a 28.
stanalger
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If "clumping" were allowed, we'd have a much longer list.
The longer list begins:
0, 1, 8, 10, 17, 18, 26, 27, 28, 45, 55, 62, 63, 71, 72, 81, 82, 89, 91, 99, 100, 107, 108, 109, 116, 125, 134, 136, 143, 144, 145, 153, 154, 161, 181, 188, 189, 197, 198, 199, 206, 207, 208, 215, 216, 224, 226, 233, 234, 235, 242, 243, 244, 253, 261, 262, 269,...

A few examples:
10^3=1000 and 10+0+0=10
99^3=970299 and 9+70+2+9+9 = 99
161^3=4173281 and 4+1+73+2+81 = 161
269^3=19465109 and 194+65+1+0+9=269
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