World!Of
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[ July 28, 2013 ]
Innovative Cashier
The topic began in WONplate 179 and continues here
B.S. Rangaswamy (email)

Go to WONplate 179

 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.

For reference goals and easy searching all the nine- & pandigitals
implicitly displayed in these topics are listed here.

Topic A.Pandigitals → 7651890243, 1048297653, 4037158926, 5498162037,
5107289346, 6924381507, 7462891053, 8172395604, 8702194356

A000184 Prime Curios! Prime Puzzle
Wikipedia 184 Le nombre 184
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