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Subject: "Pyramid"     Previous Topic | Next Topic
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Conferences The CTK Exchange High school Topic #252
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Vladimir
Member since Jun-22-03
 Jul-22-03, 09:24 AM (EST)
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"Pyramid"
 
   LAST EDITED ON Jul-22-03 AT 02:45 PM (EST)
 
9 * 1 + 2 = 11
9 * 12 + 3 = 111
9 * 123 + 4 = 1111
9 * 1234 + 5 = 11111
9 * 12345 + 6 = 111111
9 * 123456 + 7 = 1111111
9 * 1234567 + 8 = 11111111
9 * 12345678 + 9 = 111111111
9 * 123456789 + 10 = 1111111111

I ran out of digits. But wait, there's more digits in the hexadecimal system:

F * 116 + 2 = 1116
F * 1216 + 3 = 11116
F * 12316 + 4 = 111116
F * 123416 + 5 = 1111116
F * 1234516 + 6 = 11111116
F * 12345616 + 7 = 111111116
F * 123456716 + 8 = 1111111116
F * 1234567816 + 9 = 11111111116
F * 12345678916 + A = 111111111116
F * 123456789A16 + B = 1111111111116
F * 123456789AB16 + C = 11111111111116
F * 123456789ABC16 + D = 111111111111116
F * 123456789ABCD16 + E = 1111111111111116
F * 123456789ABCDE16 + F = 11111111111111116
F * 123456789ABCDEF16 + 10 = 111111111111111116

I ran out of digits again. Can you prove it without calculating? In any base?

P.S.
I added the last line to each pyramid. Now it holds even in the binary system, even though the "pyramid" has only one line:

1 * 1 + 102 = 112


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alexbadmin
Charter Member
1031 posts		 Jul-22-03, 11:39 PM (EST)
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1. "RE: Pyramid"
In response to message #0
 
   >I ran out of digits again. Can you prove it without
>calculating? In any base?

This is very simple by induction.


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Vladimir
Member since Jun-22-03
 Jul-23-03, 09:20 AM (EST)
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2. "RE: Pyramid"
In response to message #1
 
   I know that it is a simple induction. I saw this pyramid printed in about 30 years old newspaper when I was in junior high and it'stuck in my mind. Only later I proved it by induction in any base.

P.s. Thanks for erasing my error.


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Vladimir
Member since Jun-22-03
 Jul-24-03, 02:48 PM (EST)
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3. "RE: Pyramid"
In response to message #1
 
   LAST EDITED ON Jul-25-03 AT 07:44 PM (EST)
 
In any base b, the first line (in binary system at the same time the last line) is obviously true:

(b - 1) * 1 + 1 + 1 = b + 1 = (10)b + 1 = (11)b

Suppose that in base b the n-th line is correct and n + 1 < b is a digit in this base:

(b - 1) * (12...n)b + (n + 1) = (111...1)b (n + 1 ones)

(b - 1) S0n-1 (n - k - 1)bk + (n + 1) = S0n bk

Let's try the (n+1)-th line:

(b - 1) S0n (n - k)bk + (n + 2) =

= (b - 1) S1n (n - k)bk + (b - 1)(n + 1) + (n + 2) =

= (b - 1) S0n-1 (n - p - 1)bp+1 + b(n + 1) + 1 =

= b{(b - 1) S0n-1 (n - p - 1)bp + (n + 1)} + 1 =

(now substitute the n-th line)

= b S0n bp + 1 = S0n bp+1 + 1 = S1n+1 bk + 1 = S0n+1 bk

(b - 1) * (12...n+1)b + (n + 2) = (111...1)b (n + 2 ones)

The (n+1)-th line is then also correct.


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sfwc
Member since Jun-19-03
 Jul-25-03, 06:51 PM (EST)
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4. "RE: Pyramid"
In response to message #0
 
   LAST EDITED ON Jul-26-03 AT 06:06 AM (EST)
 
Of course this may be generalised in various ways. So for example going a little further down Pascal's triangle, we have:
1 = 1
1*F + 3 = 12
13*F + 6 = 123
136*F + A = 1234
136A*F + F = 12345
136AF*F + 15 = 123456
in hex. (and similarly in other bases)

Or using powers of 2:
01 = 01
01*99 + 02 = 0101
0102*99 + 04 = 010102
010204*99 + 08 = 01010204
01020408*99 + 16 = 0101020408
0102040816*99 + 32 = 010102040816
010204081632*99 + 64 = 01010204081632
01020408163264*99 + 128 = 0101020408163264
etc..

A rather nice complimentary pair emerges using alternate terms of the Fibbonachi sequence:
01 = 01
01 = 01
01*99 + 02 = 0101
01*99 + 03 = 0102
0102*99 + 05 = 010103
0103*99 + 08 = 010205
010205*99 + 13 = 01010308
010308*99 + 21 = 01020513
01020513*99 + 34 = 0101030821
01030821*99 + 55 = 0102051334
0102051334*99 + 89 = 010103082155
0103082155*99 + 144 = 010205133489
etc..

and of course the trivial:
1 = 1
1*9 + 1 = 10
11*9 + 1 = 100
111*9 + 1 = 1000
1111*9 + 1 = 10000
11111*9 + 1 = 100000
etc..
which is what the whole lot stems from

Thankyou

sfwc
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