NUMBER GAMES
BY ROBERT
T. KUROSAKA
Bob teaches
some familiar old tricks to a
new machine
Here's an old number tnck
StcHI with any four-d1g1t
number. s y. 3917 Form a
new number by arranging its
d1g1ts m decreasmg order.
9731. and another w1th the
dJgJts in increasing ord r.
13 79 Fmd the difference of
these numbers. 83 52 Repeat the procedur with 8352 8532 - 2358 ~
617 4. Another repetitiOn produces
7641- ! 467~6174 W c;eethatfurther repeti ions will yield an endless
sequence of 6174s
Que tions anse immediately Will all
four-dig1t numbers eventually arrive at
this or some other sel -replicating
value? Which numbers require the
most steps before they are repeated?
It turns out that all four-digtt numbers
t xcept multtples of II Ill wtll r ach
617 4 in at mo t seven steps.
Many similar procedures exist for
generatmg repeating sequences. and
some of them produce startlmg
results Computers make the job of
xploring them far easter than the
manual method.
Whats the purpose of this intellectual thumb-twtddhng you ask. Hardly ever a practical one I admit. but
after all the name of thts column is
" M thematical Recreations"
ThL month Ill look at several number-sequence games. providing BASIC
programs that check them out (Readers are also encouraged to seek ways
of improving on my solution l
In all cases. the term numbrr will
mean pos1tiw.> integer
SUM OF CUBES
Star w1th any number say 3 52 Find
the sum of the cubes of its digtts 31 +
r;1
21 - 160. Repeat the procedure
With 160 P + 61 +- 01
217 which
m turn g1ves 21 + 11 + 71 - 352. We
already find ourselves in a loop (160.
217, 352).
Listing I is a program that performs
the sum-of-cubes routine on any set
of consecutive numbers. For a particular 11 . it prints the consecutive
sums of cubes until a loop is found.
Readers may wish to modify the program later. perhaps suppressing the
actual terms and printmg out only the
number of steps taken and the type
of loop
It has been proved that all numbers
will eventually reach one of several
loops I have seen the proof m a Mathematical AssoCiation of America journal. but I cannot find ~hat article. nor
do I remember the proof.
The following are a few of the
known loops three-step: (55. 2 50.
133). two-step (136. 244) and (919.
1459): one-step· (153). (370). (371).
and. 0f course (II .
After exploring the sum-of-cubes
procedure. readers may wish to investigdte other powers or even other
number bases.
fACTORIAL SUMS
Start w1th any number. say, 169 Find
the um of the factorials of its d1g1ts: 11
+ 6! + 9! - 363601. Repeat the process 3! + 6' + 31 + 6' + 0' + 1! 1454 Then. I!+ 41 + 5! + 4! = 169.
and we have already found a loop.
(169, 363601. 1454)
I have examined all numbers less
than 1000 and found that nearly all
of them enter this particular loop You
will 1lso find some one-step loops.
since I -= I' 2 - 2! . and 145 ~ I!+
4' -r 5! Some two-step loops are (871.
45361) and (872 45362).
Listing 2 calculates the sum
of factorials for any range of
numbers. When numbers are
set for lo-d1git precision. the
program can handle starting
values up to 9999999999.
But even this giant yields 10
x 9! - 3628800. As long as
we limit the argument to 10 d1gits at
most. no result will exceed 7 digits. Of
course. if double precision is available.
the program can be adapted to work
with even larger numbers.
PALINDROMIC SUMS
It has been said that the first man introduced himself to the first woman
with a palindrome: "Madam. I'm
Adam!' Not to be outdone. she
replied in kind. "Eve:·
Now we'll go searching for sequences that produce numeric palindromes. that is. numbers that read the
same forward and backward.
Start with any number. say. 4e
Reverse its d1gits and add the result
to the anginal number: 48 + 84 132. Repeat the procedure: 132 + 231
- 363. a palindrome.
All numbers may eventually produce palindromes using this reverseand-add routine-but that remams to
be proved. Some numbers are known
to require many steps: for example. 89
requires 24 steps.
Charles W. 1tigg, presently book
review editor for the Journal of Recreational Mathematrcs and a prolific contributor to numerous mathematics
publications. found only 249 numbers
less than I 0.000 that require more
than I 00 steps to reach a palindrome.
Among them is the dreaded 196.
which has been taken to hundreds of
(contmutd)
Robert T Kurosaka teaches mathematics in
the Massachusetts State CDllege system. He
can be reached do BYTE. One Phoenix Mill
Lane. Peterborough. NH 034 58.
AUGUST 1986 • BYTE
313
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Sullt l"
81\d • bstlab, OH
C.ll (216) 9,. ·Y12l
Cun~J
~
thot4St~ttds of steps with no palindr o me
in s1ght. The same 1s true for it
"cousms." 295 and 394
Listing 3 investigates palindromic
sums for any range of numbers The
lower limit for the sequence is II. and
there is no explicit upper hmit. It d 'S
not have safeguards aoain t dubiou
numbers like 196 that produce very
k)ng sequence'> Improvement.., arl'
left to the readers ingenUit)
DELETE AND ADD
One of my favonte is the delightful
little marvel shown in the sample run
of hsting 4. Begin by listing the whole
numbe1 I through 11 You then delete
(u>I1 1J11Ut'dl
Listing I a: Tfte BASIC program to proJtiCl' cube sums 11111il a l!Jdl' rs frwtd
10 REM Sum of Cubes
20 REM by Bob Kurosoka
30 REM ---------------------------------------------------40 DIM A(10),8(100) :REH A() holds digits, B() holds sums .
50 CLS
60 PRINT "Program generates sequences of cub e sums.•
70 PRINT
80PRINT "Lower limit, upper limit";
90 INPUT LL, UL
100 LL•ABS(INT(LL))
110 UL•ABS(INT(UL))
120 REM ---------------------------------------------------130 fOR N•LL TO UL :REM Sequences for each no. LL to NN
140 SP•0
:REM SP coun t s the steps before o cycle
150 B(SP)•N
: REM Make the f irst term=N
160 PRINT B(SP);
: REM Print the current term
: REM Make o copy of latest term
170 M•B(SP)
180 REM --------------------------------------------------·-190 REM Break up the term into its comp onent digits
200 D•1
:REM D =no. of digits
210 T•INT(M/10)
: REM T • no. of "Tens" in M
220 A(D)=M-10•T
:REM Store rightmost digit in array A
230 If T<>0 THEN D=D+1: M=T: GOTO 10
240 REM ---------------------------------------------------250 REM Colculote the sum of the cubes of the digits in A( )
260 SUM=0
270 fOR I=1 TO D
280 SUM=SUM+A(I)•A(I)•A(I)
290 NEXT I
300 REM See if sum hos occurred already.
310 I•0
320 WHILE B(I)<>SUM AND I<=SP
330 I•I +1
340 WEND
350 If B(I)•SUM AND I<=SP THEN 400
:REM one more step
360 SP•SP+1
370 B(SP) .. SUM
: REM Store SUM in array B
380 GOTO 160
390 REM ---------------------------------------------------400 PRINT SUM; "* loops to step "; I
410 NEXT N
420 END
Listing I b : A samplt' run
Program generates sequences of cube sums .
Lower limi t , upper I l mi t 7 21 , 25
21 9 729 1080 513 153 153 • loops to step 5
22 16 217 352 160 217 • loops to step 2
23 35 152 134 92 737 713 371 371 • loops to step
24 72 351 153 153 • loops to step 3
25 133 55 250 133 • loops to step 1
Ok
7
NUMBER GAMES
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Listing 2a: Thr BASIC pro<Jram to produce fattorial sums until a cycle
IS
fotmd.
10 REM Sum of Factorials
20 REM by Bob Kurosoko
30 REM ----------------------------------------------------40 DIM A(10),B(100),F(10)
50 DATA 1,1,2,6,24,120,720,5040 , 40320,362880
60 FOR J=1 TO 10
70 READ F(J)
80 NEXT J
90 CLS
100 PRINT "Program generates factorial sum sequences."
110 PRINT
120 PRINT "Lower I imi t, upper I imi t ";
130 INPUT LL, UL
140 LL=ABS(IN~(LL))
150 UL=ABS(INT(UL))
160 REM ----------------------------------------------------170 FOR N=LL TO UL
:REM Sequences for each no. LL to NN
180 SP=0
:REM SP counts the steps before a cycle
190 B(SP)=N
:REM Make the first term=N
200 PRINT B(SP);
:REM Print the current term
210 M=B(SP)
:REM Make a copy of latest term
220 REM ----------------------------------------------------230 REM Break up the term into its component digits
240 D=1
:REM D = no. of digi ts
250 T•INT(M/10)
:REM T = no. of "Tens" in M
260 A(D)=M-10•T
:REM Store rightmost digit in array A
270 IF T<>0 THEN D=D+1: M=T: GOTO 250
280 REM ----------------------------------------------------290 REM Calculate the sum of the factorials of digits in A()
300 SUM=0
310 FOR I=1 TO D
320 SUM=SUM+F(A(l)+1)
330 NEXT I
340 REM See i f sum has occurred already.
350 I=0
360 WHILE B(l)<>SUM AND I<=SP
370 I=I+1
380 WEND
390 IF B(I)=SUM AND I<=SP THEN 440
400 SP,.SP+1
:REM one more step
410 B(SP)=SUM
: REM Store SUM in array B
420 GOTO 200
430 REM ----------------------------------------------------440 PRINT SUM; "* I oops to step "; I
450 PRINT
460 NEXT N
470 END
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Inquiry 135
64K•12BK•256K
DRAMS
Listing 2b A -amplt•
11111
Program generates factoria l sum sequences.
Lower I imit, upper I imit ? 24, 27
24 26 722 5044 169 363601 1454
25 122 5 120 4 24
169 * loops to step 9
26
722
5044
27 5042 147
to step 5
Ok
26
169
363601
5065
961
722
1454
363601
5044
169
* loops to step
169
363601
169
363601
BITTNER
3E:
1454
169 * loops to step
1454
4
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Listing 3a: Tfte BASIC pro~ram to /illd palindromrs by addim1 a •wmt•cr to its
reverse rmage <llld repeatinq lire rr,JCt'SS 1dlll tltc restiltilly sum
Operataon 11 tutty automat.c w, tft no software required
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ELEXOR
Inq uiry 103
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10 REM Palindromic Sums
20 REM by Bob Kurosoko
30 REM ------------ ----- --------------- - - - ----------- ----- -40 SP$=" "
:REM One spa ce i nside quotes
50 D$="0123456789"
60 YES•( 1=1)
: REM A( ) holds the d i g i ts.
70 DIM A(100)
80 CLS
90 PRINT"Progrom generates sequen ces ending in palindromes ."
100 PRINT
110 PRINT "Lower I im i t (>10 ) and up per I i mit ";
120 INPUT LL, UL
130 LL•ABS(INT(LL))
140 UL•ABS(INT(UL))
150 IF LL<10 THEN 110
160 FOR N=LL TO UL
170 SP•0
:REM SP cou nts th e s teps before a cyc l e
180 REM ------- ------------ --- - -------- ----------- ---------190 REM Break up the ter m int o its component digits
200 M=N
:REM Make a copy of latest t erm
210 0=1
:REM D = no. of d igits
220 T•INT(M/10)
: REM T = no. of "Te ns" in M
230 A(D)•M- 10• T
: REM Sto re rightmost d i git in array A
240 IF T<>0 THEN 0=0+1: M=T: GOTO 220
250 ODD•ABS((INT(D/2)<>0/2)) : REM Eve n or odd no . of dig i t s?
260 REM --- ----- ----- - - ------------------------------------270 REM Pr i nt the latest t er m
280 FOR I=U TO 1 STEP - 1
290 PRINT MI0$(0$ , A(l)+1 , 1);
300 NEXT I
310 PRINT SP$;
320 REM - ----------- ---- - ---- --- --- - - ------------------- - --330 REM Check for palindrome
340 FOR I•1 TO 0/2
350 PL•(A(I) =A(0-I+1))
360 IF NOT PL THEN I=0/2 : REM Exit i f no t a po l i ndrome
370 NEXT I
380 IF PL THEN 580
390 REM -------- --------------------------------- ---- ------400 REM Add each digit to its reve r s e image counterpar t
410 FOR I• 1 TO 0/2+000
420 A(I)=A( I )+A(0-1+1)
430 A(D-I+1)=A(I)
440 NEXT I
450 REM Ch~ck for corry
460 FOR 1•1 TO D
470 IF A(I)<10 THEN 500
480 A(I)•A(I)-10
490 A(I+1)=A(I+1)+1
500 NEXT I
510 IF A(D+1)=0 THEN 540
s20 o.. D+1
530 OOO=ABS((INT(0/2)<>0/ 2))
540 SP•SP+1
550 GOTO 280
560 REM ------ ------ -------------- ---------- - ------------ - - 570 REM Indicate that a cyc le ha s been found
580 PRINT "* at step " ; SP
590 FOR 1=1 TO 0
600 A(I) .. a
610 NEXT I
620 NEXT N
630 END
NUMBER GAMES
Li ting 3b. A samplt•
lUll.
.... ...::::-u.·:.c:
Program generates sequences ending in pol indromes.
·=-~---=:.-:::e
. . . . . . . . . CM&eiiK . . . .,......e ... ecelltt W0 1010011110ftQ
=--.-y~....:.:-~,."'*' · ~Hall
Lower limit (>10) ond upper I imit? 75, 80
..
....
•,_.AID....,..1501C~- IIIII . Iilvdlflier..... ~
~--· . . . snt~ . . . . . . . . . . . . ........
&c:-.Uit fFT...,.....
. _ . . _ , ,. . NO
75 132 363 • ot
76 143 484 • ot
77 • ot step 0
78 165 726 1353
79 176 847 1595
80 88 * ot step
step
step
..,.....,...,Iflll...,. ant
•........,...., ....... H.....,.,._.,._.._ 24 ... WO...,....• Cloll ....... ....,...,_,
....,,
.......
. .......
.,_, ......,............
.....,....,Oft_,....,_
.....
l tl
2
2
4884 • ot step 4
7546 14003 44044 * ot step
1
•tiMAID ......
1Dwr~1CIK
~
.....
~
ltV81~......,,
,,....,~
....... ............ ..
• _..,.. ........ c... ........ Faroe...,_. ...... ...,
6
~~·
.,~.._..c.IIIH
... C..C
~
. c.;..,,...
~~:.:.::::~ · -:..'""i:::i:L:.'='
.. ., .........
~
•ua.,....... • ....,..,....., ...... ,.....
..,JIC
m
Ok
Listmg 4a: Tlie BASIC program to perform the delete-a11d-add routi11e for any
mtcqra/
r (l\tt'r
10 REM Delete ond Add
20 REM by Bob Kurosoko
30 REM ----------------------------------------------------40 N•50
:REM N~number of integers in starting sequence.
50 DIM SF$(3), A(N) :REM SF$() = suffixes, A() =sequence.
60 DATA nd, rd, th
70 FOR Ja1 TO 3: READ SF$(J): NEXT J
80 REM ----------------------------------------------------90 CLS
100 PRINT "Program uses o delete-and-odd process"
110 PRINT "to generate iAp for i=1 to ";N; "/ p."
120 PRINT: INPUT "Enter o value for p (power)"; P
130 PsABS(INT(P))
140 IF P<2 THEN 100
150 REM ---------------- ------------------------------------160 PRINT: PRINT "Starting sequence:"
170 fOR I•1 TO N
180 A(I)•I
190 PRINT I;
200 NEXT I
210 PRINT
220 REM ---------------------------- ------------------------230 FOR R=P TO 2 STEP-1
240 DC=0
:REM Counts the terms deleted.
250 FOR J=R TO N STEP R :REM Delete every Jth term
260 A(J)=0
270 DC•DC+1
280 NEXT J
290 REM ----------------------------------------------------300 REM Print deleted orroy
310 WSF=SGN(R-3)+2
:REM Select suffix
320 PRINT: PRINT "Delete every "; R; SF$(WSF); " term:"
330 FOR I=1 TO N
340 IF A(I)<>0 THEN PRINT A(l); ELSE PRINT "•";
350 NEXT I
360 REM ----------------------------------------------------370 REM Compute Portio! Sums
380 Ka1
390 FOR J=2 TO N-DC :REM There wi I I be N-OC valid numbers.
400 K•K+1
:REM K points to next term to be added.
410 IF A(K )=0 THEN K=K+1
:REM Skip zero (deleted) terms.
420 A(J)=A (J-1)+A(K) :REM Calculate portio! sum.
430 NEXT J
440 N=N-DC
:REM Revise the number of vali d terms in A().
450 REM ------------------------------------------------- ---460 REM Print Portio! Sums
470 PRINT: PRINT: PRINT "Portio! sums:"
!CO II !HIUtd)
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Single Board Computer
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Inquiry 246
NUMBER GAMES
480
490
500
510
520
530
The delete-and-add
process produces
a power sequence.
FOR 1•1 TO N
PRINT A(l);
NEXT I
PRINT
NEXT R
END
every fourth term and list the partral
urn of the remaining terms (5e
frgure I for an explanatronl
O\\
delete every thrrd term and agam ~~~
th partral sums Then del 't • ·v ·ry
second term and list the partral sums
one last time.
We stop at this pomt (we wouldn t
delete evern term would we?l and n e
the remarkable results The equence
consrsts of the fourth powers of the
whole numbers
This procedure generalizes to nth
powers (begin by deleting every nth
term. and st on I It was drscovered by
Alfred Moessner m 1951 and prov ·d
by Oskar Perron in the same year A
thorough treatment can be found m
.. On the Moessner Theorem on Integral Powers.. by C T. Long 111 the
American Matfremalit 1/ 1ontfJ/u . vol 71
( 1966) pages 846-8 51
Listing 4 performs the above dele
and-add routine for any integral
power For larger powers. the mitral list
of vhole numbers should be xtended in lme 40. (Hint: For the nth
power. only I nth of the lis will remain
to the conclusron ) For very large
powers. the use of ext<.;nded or double precisron may be in order.
I hope you find these proccdure'i
and their results as en1oyable as I do
I would welcome any references and
sources on these or other number
sequences. •
Listing 4b: A sample run.
Program uses o delete-and-odd proces s
to generate 1•p for 1•1 to 50 I p.
Enter o value for p (power)? 4
Starting sequence :
1 2 3 4 5 6 7 8 9 10
18 19 20 21 22 23 24 25
33 34 35 36 37 38 39 40
48 49 50
Delete every
1 2 3 • 5
21 22 23 •
39 • 41 42
11
26
41
12
27
42
13
28
43
14
29
44
15
30
45
16
31
46
17
32
47
4 th term:
6 7 • 9 10 11 • 13 14 15 • 17 18 19 •
25 26 27 • 29 30 31 • 33 34 35 • 37 38
43 • 45 46 47 • 49 50
Partia l sums:
1 3 6 11 17 24 33 43
171 193 216 241 267 294
523 561 600 641 683 726
54 67 81 96 113 131 150
323 353 384 41 7 45 1 486
771 817 864 913 963
Delete every 3 rd term:
1 3. 11 17. 33 43. 67 81 • 113 131 • 17 1 193. 24 1
267 • 323 353 • 417 451 • 523 561 • 64 1 683 • 77 1 817
• 913 963
Partial sums:
1 4 1s 32 65 108
1372 1695 2048 2465
6912 7825 8788
175 256 369 see 671 864
2g16 3439 4000 4641 5324
11 e5
6095
Delet e every 2 nd term:
1 • 15. 65. 175. 369 . 671 • 1105 . 1695. 2465 . 3439 .
4641 • 6095 • 7825 •
Partial sums :
1 16 81 256
20736 28561
625
1296
2401
4096
656 1
10000
14641
Ok
Senes with fNery fourth
term deleted
New senes composed
of partial sums
1
+...-/12
1/ +3/
+../'13
. +6/
+~5
tl/
+~6
1+7/
+
~7
2.4/
+
~9
...
3t3
Figure I : In the delete-and-add program. partial sums are taken to produce a new sequence of terms Tfte {trst term is unchanged
The second term is added to the first term to produce a new second term. Tftat new term rs added to the third term to produce a new
third term. and so forth.
lJI
BY TE • AUGUST 1986