World!Of
Numbers

WON plate
184 |

[ July 28, 2013 ]
Innovative Cashier.
The search begun in WONplate 179 continues here !
by B.S. Rangaswamy (email)

 Innovative Cashier     Indian currency is available in denominations of Rupees 1, 2, 5, 10, 20, 50, 100, 500, 1000 --. In order to reduce this inventory, a Bank Cashier thinks and imagines of only two denominations of 4 & 9 rupees instead of the entire lot. With this combination Each and Every amount can be issued/received, barring the following values :

 2 6 11 19 3 7 14 23 5 10 15

However above meagre requirement can be met
with the availability of coins of denomination of Rs
1, 2, 5 & 10. With the availability of currency of
denominations Rs 400 & 900, it is possible to
reduce the bulk of currency. This innovative change
can reduce inventory to only Four - 4, 9, 400 & 900.
This innovative choice of Cashier is only
imaginary, but thought provoking, educating,
entertaining to originate the thoughts of possible
implementation.
Inspired by the cashier’s choice, I make a revised
statement that Every number above 23 is a sum of :

A. Various powers of 2 or
B. Various powers of 3 or
C. Combination of powers of 2 & 3

and is evident from the following { Note: underlined are coins }.

996 = 29 + 28 + 27 + 26 + 25 + 22 or
36 + 35 + 24 + 23 or
 4*35 + 3*23
[ 1*900 + 10*9 + 5 + 1 ]

997 = 29 + 28 + 27 + 26 + 24 + 23 + 22 + 32 or
36 + 35 + 24 + 32
[ 1*900 + 10*9 + 5 + 2 ]

998 = 29 + 28 + 27 + 26 + 24 + 32 + 32 + 22 or
36 + 35 + 32 + 32 + 23 or
 36 + 28 + 32 + 22 or 29 + 2*35
[ 1*900 + 10*9 + 2*4 ]

999 = 29 + 28 + 27 + 26 + 33 + 23 + 22 or
36 + 35 + 33
[ 1*900 + 11*9 ]

Curios Climax

by B.S. Rangaswamy [ December 2013 ]

A. Pandigitals

Pandigitals are ten digit numbers having all numerals from 0 to 9.
There exist over 3.2 million (9*9-factorial to be precise) pandigitals.
Out of these #834 numbers are gifted with the rare quality of being
the product of two 5 digit factors, which together have all numerals
from 0 to 9 as in:

76518 * 90243 = 6905213874
(all nos from 0 to 9) = (all nos from 0 to 9)

Following few such pandigital & factor combinations are taken
as curios for resolving them as sums of powers of 2 and/or 3:

10482 * 97653 = 1023598746 Lowest
40371 * 58926 = 2378901546
54981 * 62037 = 3410856297
51072 * 89346 = 4563078912
69243 * 81507 = 5643789201
74628 * 91053 = 6795103284
81723 * 95604 = 7813045692
87021 * 94356 = 8210953476 Highest

List Of #779 of such Scintillating Equations are listed out in the
book "Wonders of Numerals". Fifty five left_outs were discovered
by Patrick De Geest using his computer programming skills!

Curio 5001

1023598746 =
229 + 318 + 226 + 224 + 315 + 220 + 214 + 38 + 36 + 34 + 33 11_Tier
[ 1137331*900 + 94*9 ]

Curio 5002

2378901456 =
231 + 227 + 226 + 224 + 223 + 314 + 217 + 213 + 37 + 36 + 35 11_Tier
[ 2643223*900 + 84*9 ]

Curio 5003

3410856297 =
231 + 319 + 226 + 225 + 218 + 311 + 213 + 35 + 27 + 25 10_Tier
[ 3789840*900 + 33*9 ]

Curio 5004

4563078912 =
232 + 227 + 317 + 222 + 219 + 215 + 37 + 34 + 34 + 24 10_Tier
[ 5070087*900 + 68*9 ]

Curio 5005

5643789201 =
232 + 319 + 227 + 316 + 223 + 312 + 218 + 216 +
215 + 213 + 212 + 211 + 210 + 27 + 22 16_Tier
[ 6270876*900 + 89*9 ]

Curio 5006

6795103284 =
232 + 231 + 228 + 226 + 224 + 218 + 216 + 37 + 36 + 27 + 26 + 24 12_Tier
[ 7550114*900 + 76*9 ]

Curio 5007

7813045692 =
232 + 320 + 224 + 315 + 217 + 215 + 211 + 210 + 29 + 28 + 27 + 26 12_Tier
[ 8681161*900 + 88*9 ]

Curio 5008

8210953476 =
232 + 320 + 318 + 225 + 314 + 221 + 220 + 218 +
215 + 37 + 36 + 35 + 34 + 32 14_Tier
[ 9123281*900 + 64*9 ]

B. Pandigital Squares

There are only three pandigital squares,
whose square roots are 5 digit palindromes:

358532 = 1285437609
846482 = 7165283904
977792 = 9560732841

All these pandigitals are illustrated as sums of powers of 2 and/or 3 as below:

Curio 5009

1285437609 =
319 + 226 + 316 + 223 + 222 + 218 + 217 + 215 +
213 + 37 + 36 + 29 + 25 + 32 14_Tier
[ 1428264*900 + 9 ]

Curio 5010

7165283904 =
232 + 231 + 229 + 227 + 316 + 223 + 218 + 215 +
38 + 38 + 36 + 27 + 26 + 25 + 22 15_Tier
[ 7961426*900 + 56*9 ]

Curio 5011

9560732841 =
233 + 229 + 318 + 316 + 221 + 220 + 218 + 215 +
214 + 211 + 210 + 33 + 22 13_Tier
[ 10623036*900 + 49*9 ]

C. Elevendigital Numbers

Elevendigital numbers are 11 digit numbers
having all numerals from zero to 9.

There are only six cubes in elevendigital numbers:

23263 = 12584301976
25353 = 16290480375
27953 = 21834609875
31233 = 30459021867
35063 = 43095878216
39093 = 59730618429

Each of these elevendigital cube numbers are
illustrated as sum of powers of 2 and/or 3 as below:

Curio 5012

12584301976 =
321 + 319 + 229 + 228 + 227 + 224 + 314 + 312 +
216 + 212 + 210 + 29 + 28 + 27 + 25 15_Tier
[ 13982557*900 + 169*4 ]

Curio 5013

16290480375 =
321 + 232 + 319 + 228 + 226 + 225 + 221 + 313 +
216 + 215 + 38 + 37 + 210 + 34 + 24 + 32 16_Tier
[ 18100533*900 + 75*9 ]

Curio 5014

21834609875 =
234 + 232 + 228 + 226 + 224 + 314 + 221 + 312 +
215 + 38 + 29 + 28 + 27 + 26 + 23 15_Tier
[ 24260677*900 + 63*9 + 2*4 ]

Curio 5015

30459021867 =
234 + 321 + 231 + 229 + 227 + 311 + 215 + 214 + 36 + 27 + 25 + 22 11_Tier
[ 33843357*900 + 63*9 ]

Curio 5016

43095878216 =
235 + 233 + 227 + 223 + 221 + 220 + 218 + 217 +
310 + 29 + 28 + 27 + 33 + 22 14_Tier
[ 47884309*900 + 12*9 + 2*4 ]

Curio 5017

59730618429 =
235 + 234 + 232 + 320 + 318 + 224 + 314 + 218 +
213 + 38 + 210 + 29 + 26 + 32 14_Tier
[ 66367353*900 + 81*9 ]

D. Palindromic Cube Number

It is astonishing to learn that a 11 digit number
is a cube and also a palindrome:

22013 = 10662526601

This is the one and only palindrome cube whose root is nonpalindromic.
It was first noticed by Trigg in the year 1961 (by me in 2001).
Search for a second such palindromic cube have failed so far.

Curio 5018

10662526601 =
321 + 227 + 226 + 312 + 218 + 215 + 39 + 36 + 25 + 32 10_Tier
[ 11847251*900 + 77*9 + 2*4 ]

E. Even Digit Squares

Even digit palindromic squares are very scarce.
Lowest even digit palindromic square is:

8362 = 698896

Next one is

7986442 = 637832238736

which is now the intended curio of 12 digits,
for coining its constituent powers of 2 and/or 3.

Curio 5019

637832238736 =
239 + 236 + 234 + 231 + 224 + 223 + 222 + 311 +
215 + 39 + 212 + 36 + 36 13_Tier
[ 708702487*900 + 48*9 + 4 ]

Highest known even digit palindrome square is of 52 digits,
which was discovered by Pete Leadbetter of England
on 20th May 2001, after a 23 day continuous search!

F. Giant 24 Digital

Curio 5022

10^23 =
348 + 274 + 270 + 267 + 263 + 262 + 260 + 259 +
258 + 256 + 249 + 246 + 245 + 244 + 326 + 240 +
324 + 235 + 232 + 319 + 229 + 224 + 314 + 220 +
218 + 216 + 213 + 212 + 25 + 24 + 32 + 23 32_Tier

[111111111111111111111*900 + 25*4]

G. Giant 33 Digital

Curio 5023

10^32 =
367 + 2102 + 2100 + 299 + 298 + 290 + 355 + 286 +
285 + 283 + 352 + 280 + 348 + 347 + 272 + 270 +
269 + 343 + 265 + 264 + 263 + 262 + 259 + 256 +
252 + 251 + 250 + 249 + 244 + 242 + 238 + 323 +
235 + 232 + 230 + 226 + 221 + 312 + 216 + 37 +
210 + 29 + 28 + 33 44_Tier

[111111111111111111111111111111*900 + 25*4]

Details of multiples of 900, 400, 9 and 4 indicated in
brackets against each curio is the dream of
Innovative Cashier, which remains to be fullfilled.

A000184 Prime Curios! Prime Puzzle
Wikipedia 184 Le nombre 184
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