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gordonm888
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February 23rd, 2021 at 1:19:10 PM permalink
Several years ago I started analyzing the frequency with which words appeared in Boggle, and ever since I have been playing around with the combination math involved in placing strings of characters into grids of finite size, i.e. grids with edges. The puzzle below, which is one of the simplest problems of this kind, will serve to illustrate what this means.

Given a 3x3 grid of 9 squares; and each square is considered to be connected to other squares in the grid by their face or by their corners, as in the game Boggle. Thus, the center square is connected to the other 8 squares, corner squares are connected to 3 squares and edge squares are connected to 5 squares.

Now consider a string of 3 characters. Let's make the characters letters and the string a 3-letter word, to continue to emulate the game Boggle. When placing a 3-letter word into the 3x3 grid the rules are:

1. The other six squares will be blank
2. The string of letters must be arranged in the grid on connected squares such that they can be read in the correct sequence to form the word.
3. No square in the grid can be used twice in “spelling” the word.
4. For a word that is a palindrome such as “WOW” the fact that it can be read forwards and backwards does not mean that it can be counted twice; any distinct arrangement of a palindromic word counts as one single arrangement.
Note: Rules 2-4 are standard Boggle rules.

How many distinct arrangements of the following three letter words can be formed in a 3x3 grid of 9 squares?
SEX
EEL
OFF
WOW
ZZZ
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
gordonm888
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February 23rd, 2021 at 6:43:34 PM permalink
I think I killed the thread.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
chevy
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February 23rd, 2021 at 6:51:50 PM permalink
Quote: gordonm888



How many distinct arrangements of the following three letter words can be formed in a 3x3 grid of 9 squares?
SEX
EEL
OFF
WOW
ZZZ



I admit to not understanding what you are asking. I count at least 13 letters needed (SEXELOFFWWZZZ) to form the 5 words, but only a 3x3=9 grid.

Obviously I am missing something (After I get past that hurdle, then I can fall back on just not being able to solve)
chevy
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February 23rd, 2021 at 7:23:49 PM permalink

Unless??????????
Are we supposed to fit one word into the 3x3 grid in as many ways as possible

For example SEX

S corner : 15 ways * 4 corners = 60
S middle of side : 19 ways * 4 middles = 76
S center : 16 ways

Total = 152

Or if orientation of the board does not matter
SEX on top

same as

X
E
S on side


Then 15 + 19 + 4 = 38

?????????????


Then repeat for the other words?
ThatDonGuy
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gordonm888
February 23rd, 2021 at 7:26:19 PM permalink
Quote: chevy

Quote: gordonm888



How many distinct arrangements of the following three letter words can be formed in a 3x3 grid of 9 squares?
SEX
EEL
OFF
WOW
ZZZ



I admit to not understanding what you are asking. I count at least 13 letters needed (SEXELOFFWWZZZ) to form the 5 words, but only a 3x3=9 grid.

Obviously I am missing something (After I get past that hurdle, then I can fall back on just not being able to solve)


What I think he means is this:
How many ways can each of the three sets of letters be placed in a 3x3 grid so the second letter is adjacent to the first and the third is adjacent to the second? Each one is meant to be treated separately.

I assume that EEL and OFF have the same number, since each spelling of EEL can be changed to one of OFF by replacing the L with an O, the Es with Fs, and going in the reverse direction.
chevy
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February 23rd, 2021 at 7:41:32 PM permalink
Quote: ThatDonGuy


What I think he means is this:
How many ways can each of the three sets of letters be placed in a 3x3 grid so the second letter is adjacent to the first and the third is adjacent to the second? Each one is meant to be treated separately.

I assume that EEL and OFF have the same number, since each spelling of EEL can be changed to one of OFF by replacing the L with an O, the Es with Fs, and going in the reverse direction.



Okay, that makes sense. Two more questions


1) Does orientation matter?
SEX. (top)
vs
X
E
S
on left side
2) Palindromes ruled out....but what about same 3 squares in different order?
Squares 1,2,4

E E
L

Form EEL by 124 and 214

chevy
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February 23rd, 2021 at 8:02:19 PM permalink


If orientation and sequence matter I get the same for EEL and SEX, so I will assume they don't as it makes the problem more interesting. Listing ways with first letter in corner, side middle, and center...not counting duplicate triples.

SEX = 15 + 19 + 4 = 38 ways???
EEL = 15 + 13 + 0 = 28 ways???

OFF = 28 per ThatDonGuy's comment.

ZZZ= 10 + 2 + 0 = 12 ways ???

charliepatrick
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February 24th, 2021 at 8:18:59 AM permalink
Quote: gordonm888

...
How many distinct arrangements of the following three letter words can be formed in a 3x3 grid of 9 squares?
SEX
EEL
OFF
WOW
ZZZ

For simplicity assume the squares are:-
A B C
D E F
G H J
Then there are 84 sets of three squares. They fall into three types
(i) Linear - e.g. A-B-C - you can only go in a forwards or backwards direction as you cannot reach A from C - so ABC or CBA.
(ii) Circular - e.g. A-B-D - you can reach A from D so they form a circle (or triangle) - so ABD BDA DAB DBA BAD ADB.
(iii) Impossible - e.g. A-B G - you cannot form a joined triangle. You only need these to check the total combinations is correct. (I couldn't see a mathematical way of working out the combinations, so just listed them all. With larger numbers one might write a program to go through them, but then there would be more "shapes" than linear and circular.)

Linear
SEX/EEL/OFF - you can go forwards or backwards (2)
WOW/ZZZ - you can only go forwards (as the backwards is an identical placement of letters) (1)
Circular
SEX - where all the letters are different you can use any of the six ways (6)
EEL/OFF/WOW - where one of the letters is different it can be in any of three places (3)
ZZZ - where all the letters are different only one comination counts (1)
NumberTriple typeSEXEELOFFWOWZZZ
32
Linear
2
2
2
1
1
16
Circular
6
3
3
3
1
160
112
112
80
48

chevy
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February 24th, 2021 at 9:55:04 AM permalink


Your solution is more elegant than me. I did the problem by a more brute force approach. I think I get similar answers for EEL/OFF and ZZZ.

But if we consider SEX....I considered possibilities with S in corner, S in Middle of a side, S in center....I can't find where I come up short of your number.

using you grid labels

corner: (15 ways)
S at A, E can be B.....4 choices for X....(D,E,F,C)
S at A, E can be D.....4 choices for X....(B,E,H,G)
S at A, E can be E.....7 choices for X....(B,C,F,J,H,G,D)

middle of side : (19 ways)
S at B, E can be at A......2 choices for X.....(D,E)
S at B, E can be at C......2 choices for X.....(E,F)
S at B, E can be at D......4 choices for X.....(A,E,H,G)
S at B, E can be at F......4 choices for X.....(C,E,H,J)
S at B, E can be at E.....7 choices for X.....(A,D,G,H,J,F,C)

center : (16 ways)
S at E, E can be at the other 8 locations, and X can be either side of E (around the perimeter)

Total ways to have SEX
=4 corners + 4 middles sides + 1 center
=4*15+4*19+16
=152 ways


Anybody see where I am missing some SEX.....8 of them apparently???

charliepatrick
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February 24th, 2021 at 11:01:41 AM permalink
Quote: chevy

Anybody see where I am missing some SEX.....8 of them apparently???

If you go from E to A, then the next place can only be D or B. However if you go from E to B, then you can go to A C D or F. This adds two more for EB, and similarly for ED EF and EH.
gordonm888
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February 24th, 2021 at 3:06:58 PM permalink
Quote: chevy

Okay, that makes sense. Two more questions


1) Does orientation matter?
SEX. (top)
vs
X
E
S
on left side
2) Palindromes ruled out....but what about same 3 squares in different order?
Squares 1,2,4

E E
L

Form EEL by 124 and 214



Orientation does not matter. Words can be read from the bottom up or backwards or forwards, etc. They may be twisted into what I call a triangle, such as

SX
E

or

XE
S

or whatever.

Palindromes simply lack the ability to be sequenced backwards in a particular set of squares because that would not count as a distinct arrangement with respect to being sequenced forwards.

You are correct about off and eel having the same number of arrangements. I listed both of them just to have the complete set of symmetries for a 3-character string.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
gordonm888
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February 24th, 2021 at 3:07:01 PM permalink
Quote: chevy

Okay, that makes sense. Two more questions


1) Does orientation matter?
SEX. (top)
vs
X
E
S
on left side
2) Palindromes ruled out....but what about same 3 squares in different order?
Squares 1,2,4

E E
L

Form EEL by 124 and 214



Orientation does not matter. Words can be read from the bottom up or backwards or forwards, etc. They may be twisted into what I call a triangle, such as

SX
E

or

XE
S

or whatever.

Palindromes simply lack the ability to be sequenced backwards in a particular set of squares because that would not count as a distinct arrangement with respect to being sequenced forwards.

You are correct about off and eel having the same number of arrangements. I listed both of them just to have the complete set of symmetries for a 3-character string.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
gordonm888
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February 24th, 2021 at 3:16:42 PM permalink
Quote: charliepatrick

For simplicity assume the squares are:-
A B C
D E F
G H J
Then there are 84 sets of three squares. They fall into three types
(i) Linear - e.g. A-B-C - you can only go in a forwards or backwards direction as you cannot reach A from C - so ABC or CBA.
(ii) Circular - e.g. A-B-D - you can reach A from D so they form a circle (or triangle) - so ABD BDA DAB DBA BAD ADB.
(iii) Impossible - e.g. A-B G - you cannot form a joined triangle. You only need these to check the total combinations is correct. (I couldn't see a mathematical way of working out the combinations, so just listed them all. With larger numbers one might write a program to go through them, but then there would be more "shapes" than linear and circular.)

Linear
SEX/EEL/OFF - you can go forwards or backwards (2)
WOW/ZZZ - you can only go forwards (as the backwards is an identical placement of letters) (1)
Circular
SEX - where all the letters are different you can use any of the six ways (6)
EEL/OFF/WOW - where one of the letters is different it can be in any of three places (3)
ZZZ - where all the letters are different only one comination counts (1)
NumberTriple typeSEXEELOFFWOWZZZ
32
Linear
2
2
2
1
1
16
Circular
6
3
3
3
1
160
112
112
80
48



This is absolutely CORRECT, and using the same methodology that I use.


The methodology used by Charlie allows you to straightforwardly address how many three character strings would be on a 4x4 grid or more generally, on an n x n grid.

This methodology is more complex when you are analyzing strings that are 4, 5 and 6 characters in length because there are far more spatial arrangements that the string can be twisted into, especially for 4x4 and larger grids, Also there are far more symmetrical arrangements for repeated characters in the strings that need to be considered. I have never attempted a 7 character string.

And I really wish I was smart enough to figure out an approach for calculating how many ways a 9 character string can be arranged in a 3x3 grid. I've never really seriously attempted that.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
chevy
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February 24th, 2021 at 3:32:49 PM permalink
Quote: charliepatrick

If you go from E to A, then the next place can only be D or B. However if you go from E to B, then you can go to A C D or F. This adds two more for EB, and similarly for ED EF and EH.



Thank you very much!!! I guess I ended my counting of SEX from the center square prematurely.
charliepatrick
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February 25th, 2021 at 2:11:12 AM permalink
Quote: gordonm888

....I really wish I was smart enough to figure out an approach for calculating how many ways a 9 character string can be arranged in a 3x3 grid....

I didn't know how to do this so initially did it by hand. However when the alternative answer of 152 came in I couldn't see the 8 missing ones, so only then knocked up a program to see which answer was correct. This could easily be extended for larger grids but doesn't find nor define which ones have which patterns. It can only really be used to confirm the total number of valid strings.
(Some lines omitted - for larger grids I'd do it the other way round as there are fewer valid links than invalid ones)

n=0;
for (i=1; i<=9; i++)
{
for (j=1; j<=9; j++)
{
for (k=1; k<=9; k++)
{
invalid=0;
if (i==j) {invalid=1;};
if (i==k) {invalid=1;};
if (j==k) {invalid=1;};
/* Check first move is legal */
if ((i==1)&&(j==3)) {invalid=1;};
if ((i==1)&&(j==6)) {invalid=1;};
etc.
if ((i==9)&&(j==4)) {invalid=1;};
if ((i==9)&&(j==7)) {invalid=1;};

/* Check second move is legal */
if ((j==1)&&(k==3)) {invalid=1;};
if ((j==1)&&(k==6)) {invalid=1;};
etc.
if ((j==9)&&(k==4)) {invalid=1;};
if ((j==9)&&(k==7)) {invalid=1;};

if (invalid==0)
{
n++;
outline="N("+n+") ";
outline+="I: "+i+" ";
outline+="J: "+j+" ";
outline+="K: "+k+" ";
outline+="<BR>";
document.write(outline);
};
};
};
};
Gialmere
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February 25th, 2021 at 8:28:54 AM permalink


You don't play golf but decide to try a bizarre online system for the game that uses meditation and self hypnosis.

Before playing, you intensely concentrate on two strokes of different lengths to use for the entire round, one a drive, the other an approach, and play directly toward the hole so that various combinations of the two distances will get you there.

The ball must go the full length on each stroke, but you may go beyond the hole with either stroke, then play back toward the hole. All strokes are on a straight line toward the hole and will go in the cup if the stroke and hole distances are equal.

Using this odd technique of selecting two exact distances only, what is your lowest score possible on the following nine-hole course?


Hole Length Score
#1 150 yds ?
#2 300 yds ?
#3 250 yds ?
#4 325 yds ?
#5 275 yds ?
#6 350 yds ?
#7 225 yds ?
#8 450 yds ?
#9 425 yds ?
Have you tried 22 tonight? I said 22.
Joeman
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February 25th, 2021 at 9:20:39 AM permalink
I can get down to 28 strokes.

Similar to my actual golf game, my Drive would be 150 yd, and my Approach would be 25 yd. Out in 28:

Hole Length Score Shots
#1 150 yds 1 150
#2 300 yds 2 150 + 150
#3 250 yds 4 150 + 150 - 25 - 25
#4 325 yds 3 150 + 150 + 25
#5 275 yds 3 150 + 150 - 25
#6 350 yds 4 150 + 150 + 25 + 25
#7 225 yds 4 150 + 25 + 25 + 25
#8 450 yds 3 150 + 150 + 150
#9 425 yds 4 150 + 150 + 150 - 25
"Dealer has 'rock'... Pay 'paper!'"
charliepatrick
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Gialmere
February 25th, 2021 at 9:23:34 AM permalink
I'll have to think about an analytical approach to this but I did like my first guess....
Dividing all the lengths by 25 gets 6 12 10 13 11 14 9 17 18 = 6 17 18 (9 thru 14). 7 and 3 (i.e. 175 and 75 yards) gives 26 shots. (2 4 2 3 3 2 3 3 4).
Wizard
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February 25th, 2021 at 9:28:53 AM permalink
Chag Purim Sameach!, my fellow math enthusiasts.

Here is my guess,

30



75 and 125 yards. I can complete the course with a score of 30.

I am not saying this is optimal, but have a good feeling about it.

I first divided all the yardages by 25 to get:

6
12
10
13
11
14
9
18
17

Then I just know somewhere that primes are good for constructing other numbers. With a little trial and error 3 and 5 seem to work well.

For example,
hole 1: 3+3
hole 2: 5+5+5-3
hole 3: 5+5

Can anyone beat my 30?

"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
Wizard
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February 25th, 2021 at 9:55:42 AM permalink
Looks like Charlie is winning the WoV golf match so far and I'm losing, which is usually the case when I golf.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
ThatDonGuy
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charliepatrick
February 25th, 2021 at 11:20:03 AM permalink
Quote: Wizard

Looks like Charlie is winning the WoV golf match so far and I'm losing, which is usually the case when I golf.


I did a brute force search on every pair of driver and approach clubs in 25-yard intervals up to 500 yards, and get the same answer Charlie got.



Note that X indicates that reaching that hole is impossible
LongShort123456789Score
502536576759957
752524455636742
755024454536639
1002533444536537
10050233XX4X5XX
1007523343435532
1252524253436534
1255033244435432
1257524233434530
12510033234324428
1502512433443428
15050123XX3X3XX
1507512XXXX23XX
150100122XX3X3XX
15012512242673330
1752524435236534
1755033443224530
1757524233234326
17510043352263432
17512562275236336
17515012729253839
2002535354424333
20050232XX3X3XX
2007524432335430
200100X2XXXXXXXX
20012548223563740
200150123XX2X3XX
2001751295117236459
2252544253612330
2255035532412530
2257522XXXX12XX
22510053427312330
225125106239212540
22515012XXXX12XX
2251756121053212950
2252001273951112252
2502553142524531
25050321XX3X3XX
2507524125436330
250100231XX2X3XX
250125XX1XXXXXXX
250150121XX3X3XX
2501754813132612251
250200631XX5X2XX
2502251251739121555
2752562231436734
2755036521624433
2757524651236332
27510063581493746
2751252427110116346
275150124121973241
275175126141112921370
27520048131513612567
27522561210311412756
275250123151718151375
3002561322347634
30050312XX2X4XX
3007521XXXX23XX
300100X1XXXXXXXX
3001259121184510252
30015011XXXXX2XX
300175101491221511367
300200X1XXXXXXXX
30022541XXXX12XX
300250611XX3X7XX
3002751212031518131184
3252562413256534
3255036512736336
3257524217636536
325100736110524240
3251256122158921156
32515012111144831357
3251752410172136954
3252001594117141231186
325225146161111812372
325250481115176121983
3252756121011142118588
3253001212012231811997
3502563524165436
35050323XX1X3XX
3507524872136235
350100332XX1X2XX
3501251492513124858
350150125XX1X3XX
350175XXXXX1XXXX
350200247XX1X6XX
35022571441991121673
350250961XX1X3XX
35027548151711612367
3503006110XX1X5XX
35032512242012211897114
3752564635274340
3755036523748240
3757522XXXX32XX
37510083711261251064
375125XX2XXXXXXX
37515012XXXX23XX
3751751814817112510792
375200510167134191518107
37522524XXXX12XX
375250XX1XXXXXXX
3752751661813120112215122
37530041XXXX63XX
375325612101251423183112
375350122420262211875135
4002565746383244
40050334XX2X2XX
4007524329836744
400100X2XXXXXXXX
40012536210214159566
400150122XX5X3XX
40017591815852241477
400200XXXXXXXXXX
4002251671018214121291
400250241XX9X6XX
400275242141611191823127
400300X1XXXXXXXX
40032548261171961221114
40035061210XX1X3XX
4003751224202622281853158
4252566857492148
4255036534759143
42575241093236140
425100938213736152
425125122215116714170
42515012617213103155
42517561225192918174
425200181151481724180
4252258162213191012192
425250211512374189199
42527524229114176176
425300141411162321131104
425325186201132227241132
425350481719281612196
4253756121029271425181142
4254001224202622281831154
4502567968591253
45050345XX3X1XX
4507523XXXX31XX
450100433XX2X1XX
450125816227111211574
45015012XXXXX1XX
4501751792016221311999
450200952XX8X1XX
450225XXXXXX11XX
4502505101XX7X1XX
450275102071214151979
45030021XXXXX1XX
450325211141231816113108
45035011613XX1X1XX
45037548XXXX61XX
45040061210XX14X1XX
4504251224202622281811152
47525681079692360
4755036545769247
475752443111036952
475100103914381571382
475125181227172361178
475150121521961131877
4751751426131722116596
4752006121922259188101
475225188211514122384
47525011221192516247107
4752752418107126211223142
475300151251712471427122
475325241419261961192
475350918425111302720145
47537520622153124132617174
475400483032192161223155
4754256121031291427181148
47545012242026222818134185
500256910810793466
50050355XX4X2XX
500752412114336247
500100X3XXXXXXXX
500125XX2XXXXXXX
500150127XX2X3XX
500175361421021891377
500200X2XXXXXXXX
500225918155221121184
500250XX1XXXXXXX
500275221316271102421116
500300X1XXXXXXXX
5003253027171224122419156
500350249XX1X6XX
500375XX4XXXXXXX
500400X6XXXXXXXX
50042548192132346121137
50045061210XX14X1XX
500475122420262228183634220


Last edited by: ThatDonGuy on Feb 25, 2021
charliepatrick
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February 25th, 2021 at 11:38:05 AM permalink
Thanks - the guess was only based on an observation...
the logic that you needed a long club to get near the higher numbers and with 7/3 I could see you could easily get 10, 14 and 17.
Gialmere
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February 25th, 2021 at 5:20:56 PM permalink
Quote: charliepatrick

I'll have to think about an analytical approach to this but I did like my first guess....

Dividing all the lengths by 25 gets 6 12 10 13 11 14 9 17 18 = 6 17 18 (9 thru 14). 7 and 3 (i.e. 175 and 75 yards) gives 26 shots. (2 4 2 3 3 2 3 3 4).


Correct!!


Quote: ThatDonGuy

I did a brute force search on every pair of driver and approach clubs in 25-yard intervals up to 500 yards, and get the same answer Charlie got.



Note that X indicates that reaching that hole is impossible
LongShort123456789Score
502536576759957
752524455636742
755024454536639
1002533444536537
10050233XX4X5XX
1007523343435532
1252524253436534
1255033244435432
1257524233434530
12510033234324428
1502512433443428
15050123XX3X3XX
1507512XXXX23XX
150100122XX3X3XX
15012512242673330
1752524435236534
1755033443224530
1757524233234326
17510043352263432
17512562275236336
17515012729253839
2002535354424333
20050232XX3X3XX
2007524432335430
200100X2XXXXXXXX
20012548223563740
200150123XX2X3XX
2001751295117236459
2252544253612330
2255035532412530
2257522XXXX12XX
22510053427312330
225125106239212540
22515012XXXX12XX
2251756121053212950
2252001273951112252
2502553142524531
25050321XX3X3XX
2507524125436330
250100231XX2X3XX
250125XX1XXXXXXX
250150121XX3X3XX
2501754813132612251
250200631XX5X2XX
2502251251739121555
2752562231436734
2755036521624433
2757524651236332
27510063581493746
2751252427110116346
275150124121973241
275175126141112921370
27520048131513612567
27522561210311412756
275250123151718151375
3002561322347634
30050312XX2X4XX
3007521XXXX23XX
300100X1XXXXXXXX
3001259121184510252
30015011XXXXX2XX
300175101491221511367
300200X1XXXXXXXX
30022541XXXX12XX
300250611XX3X7XX
3002751212031518131184
3252562413256534
3255036512736336
3257524217636536
325100736110524240
3251256122158921156
32515012111144831357
3251752410172136954
3252001594117141231186
325225146161111812372
325250481115176121983
3252756121011142118588
3253001212012231811997
3502563524165436
35050323XX1X3XX
3507524872136235
350100332XX1X2XX
3501251492513124858
350150125XX1X3XX
350175XXXXX1XXXX
350200247XX1X6XX
35022571441991121673
350250961XX1X3XX
35027548151711612367
3503006110XX1X5XX
35032512242012211897114
3752564635274340
3755036523748240
3757522XXXX32XX
37510083711261251064
375125XX2XXXXXXX
37515012XXXX23XX
3751751814817112510792
375200510167134191518107
37522524XXXX12XX
375250XX1XXXXXXX
3752751661813120112215122
37530041XXXX63XX
375325612101251423183112
375350122420262211875135
4002565746383244
40050334XX2X2XX
4007524329836744
400100X2XXXXXXXX
40012536210214159566
400150122XX5X3XX
40017591815852241477
400200XXXXXXXXXX
4002251671018214121291
400250241XX9X6XX
400275242141611191823127
400300X1XXXXXXXX
40032548261171961221114
40035061210XX1X3XX
4003751224202622281853158
4252566857492148
4255036534759143
42575241093236140
425100938213736152
425125122215116714170
42515012617213103155
42517561225192918174
425200181151481724180
4252258162213191012192
425250211512374189199
42527524229114176176
425300141411162321131104
425325186201132227241132
425350481719281612196
4253756121029271425181142
4254001224202622281831154
4502567968591253
45050345XX3X1XX
4507523XXXX31XX
450100433XX2X1XX
450125816227111211574
45015012XXXXX1XX
4501751792016221311999
450200952XX8X1XX
450225XXXXXX11XX
4502505101XX7X1XX
450275102071214151979
45030021XXXXX1XX
450325211141231816113108
45035011613XX1X1XX
45037548XXXX61XX
45040061210XX14X1XX
4504251224202622281811152
47525681079692360
4755036545769247
475752443111036952
475100103914381571382
475125181227172361178
475150121521961131877
4751751426131722116596
4752006121922259188101
475225188211514122384
47525011221192516247107
4752752418107126211223142
475300151251712471427122
475325241419261961192
475350918425111302720145
47537520622153124132617174
475400483032192161223155
4754256121031291427181148
47545012242026222818134185
500256910810793466
50050355XX4X2XX
500752412114336247
500100X3XXXXXXXX
500125XX2XXXXXXX
500150127XX2X3XX
500175361421021891377
500200X2XXXXXXXX
500225918155221121184
500250XX1XXXXXXX
500275221316271102421116
500300X1XXXXXXXX
5003253027171224122419156
500350249XX1X6XX
500375XX4XXXXXXX
500400X6XXXXXXXX
50042548192132346121137
50045061210XX14X1XX
500475122420262228183634220



If you're looking for another round of golf...

According to the programming website I found this puzzle at, in addition to the set {175, 75}, there is another set of numbers that will complete the course in the minimum of 26 strokes. They don't post the answer but, from ThatDonGuy's list, they're not a multiples of 25 (and possibly not even whole numbers).

--------------------------------

Have you tried 22 tonight? I said 22.
ThatDonGuy
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February 25th, 2021 at 6:23:01 PM permalink
Quote: Gialmere

If you're looking for another round of golf...

According to the programming website I found this puzzle at, in addition to the set {175, 75}, there is another set of numbers that will complete the course in the minimum of 26 strokes. They don't post the answer but, from ThatDonGuy's list, they're not a multiples of 25 (and possibly not even whole numbers).


I have checked every combination of clubs that are multiples of 0.1 yards up to 900, but can't find any other than the one posted solution.
Gialmere
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February 25th, 2021 at 8:32:56 PM permalink
Quote: ThatDonGuy

Quote: Gialmere

If you're looking for another round of golf...

According to the programming website I found this puzzle at, in addition to the set {175, 75}, there is another set of numbers that will complete the course in the minimum of 26 strokes. They don't post the answer but, from ThatDonGuy's list, they're not a multiples of 25 (and possibly not even whole numbers).


I have checked every combination of clubs that are multiples of 0.1 yards up to 900, but can't find any other than the one posted solution.


Oops! My bad. I messed up one of the hole lengths. Hole #8 is only 400 yds. Fortunately the 75/175 solve works for either length but, apologies to ThatDonGuy.

The correct length does, however, make the second solve much simpler.
Have you tried 22 tonight? I said 22.
charliepatrick
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February 25th, 2021 at 9:45:38 PM permalink
Quote: Gialmere

...
Oops! My bad. I messed up one of the hole lengths. Hole #8 is only 400 yds. Fortunately the 75/175 solve works for either length but, apologies to ThatDonGuy....

Yes 150 125 works (1 2 2 4 2 5 4 3 3). 200/75 and 250/75 now get close with 27 shots.
btw I couldn't see how fractional ones could easily work as to get to 26 strokes you need at least one hole which is a 2. So if the clubs were X+fraction and Y-fraction (fraction=1/2 1/3 etc) then the only way to get whole numbers is X-Y 2X-2Y nX etc. Apart from 6 and 12 you have no others that are double.
Also you know that some combinations can't work, such as when both clubs are a multiple of 50 or where one club is a multiple of the other. However if using a brute force method that's not worth worrying about.
teliot
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February 25th, 2021 at 10:10:00 PM permalink
This very nice problem about golfing is also related to the postage stamp problem and minimizing total stamp usage across a variety of postages. Well, maybe not that related since you can't have negative stamps.

Postage Stamp Problem
Given stamps with postage x and y, relatively prime, you can always make every postage above (x - 1) * (y - 1).

I believe finding a closed form solution for the postage stamp problem for three postages is still unsolved.
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ThatDonGuy
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February 26th, 2021 at 7:15:42 AM permalink
Quote: Gialmere

Oops! My bad. I messed up one of the hole lengths. Hole #8 is only 400 yds. Fortunately the 75/175 solve works for either length but, apologies to ThatDonGuy.

The correct length does, however, make the second solve much simpler.


I checked every possible pair of different clubs that are multiples of 1/5, 1/6, 1/7, and 1/8 yard up to 1000 yards, and still get 175/75 as the only 26-stroke solution.

Update: now up through 1/14 of a yard - still only one solution for 26 (and two for 27).
Last edited by: ThatDonGuy on Feb 26, 2021
gordonm888
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February 26th, 2021 at 6:18:21 PM permalink



EDIT:
If the 8th hole is only 400 yds, rather than 450:

250 and 75 seems to yield just 27 strokes

150 =75+75_____2
300=75 +75+ 75+75__4
250 = 250 ________1
325 = 250 + 75______2
275 = 250 -75 -75 -75 +250_________5
350 = 250 -75 -75 +250_______4
225 = 75 +75 +75 _______3
400 = 250 +75 +75 _________________3
425 = 250 -75 +250 ____3



EDit: I see that ThatDonGuy has already overwhelmed the problem.
Last edited by: gordonm888 on Feb 26, 2021
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teliot
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February 26th, 2021 at 7:02:26 PM permalink
On the topic of Golfing, let me create a different problem that is based on efficient stamp totals.

Suppose the Post Office issues two values of stamps X cents and Y cents. What are the most efficient values for these two stamps X & Y so that we can make each postage from $1.20 to $2.00, using the least number of stamps overall?

Yeah, "Easy" Math Puzzles. :)
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gordonm888
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February 26th, 2021 at 7:18:25 PM permalink
Quote: teliot

On the topic of Golfing, let me create a different problem that is based on efficient stamp totals.

Suppose the Post Office issues two values of stamps X cents and Y cents. What are the most efficient values for these two stamps X & Y so that we can make each postage from $1.20 to $2.00, using the least number of stamps overall?

Yeah, "Easy" Math Puzzles. :)





My quick guess is 12 cents and 11 cents.

We need two numbers -one, call it n, that is even and the other that has multiples that are capable of making xmodn, x=1...n-1.
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teliot
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February 26th, 2021 at 9:46:02 PM permalink
Quote: gordonm888



My quick guess is 12 cents and 11 cents.

We need two numbers -one, call it n, that is even and the other that has multiples that are capable of making xmodn, x=1...n-1.

Your answer requires a total of 1118 stamps. My answer requires 1072 stamps.
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charliepatrick
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February 27th, 2021 at 1:34:02 AM permalink
Quote: teliot

...My answer requires 1072 stamps.

I think you accidentally (or not) gave a hint earlier as 120 = (13-1)*(11-1) and both 13 and 11 are primes; and they seem to work. The early values can be achieved in either 10 or 11 stamps, e.g 120 = 5*each, 121=11*11c, 122 is 4*11c+6*13c, so to get 2 more you replace one 11 with a 13. At the end 199 needs 17 stamps, and 200 (which is 208-8 so 12*13+4*11 requires 16 stamps.) The total is 1072.
teliot
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February 27th, 2021 at 6:04:09 AM permalink
Quote: charliepatrick

I think you accidentally (or not) gave a hint earlier as 120 = (13-1)*(11-1) and both 13 and 11 are primes; and they seem to work. The early values can be achieved in either 10 or 11 stamps, e.g 120 = 5*each, 121=11*11c, 122 is 4*11c+6*13c, so to get 2 more you replace one 11 with a 13. At the end 199 needs 17 stamps, and 200 (which is 208-8 so 12*13+4*11 requires 16 stamps.) The total is 1072.

Yes!

Now, what's the answer if we also need to make a postage of $1.19? This is where it gets slightly evil.


1, 2, Total = 6560
1, 3, Total = 4415
1, 4, Total = 3362
1, 5, Total = 2747
1, 6, Total = 2349
1, 7, Total = 2075
1, 8, Total = 1886
1, 9, Total = 1743
1, 10, Total = 1640
1, 11, Total = 1559
1, 12, Total = 1496
1, 13, Total = 1451
1, 14, Total = 1431
1, 15, Total = 1389
1, 16, Total = 1394
1, 17, Total = 1367
1, 18, Total = 1400
1, 19, Total = 1379
1, 20, Total = 1394
1, 21, Total = 1399
1, 22, Total = 1445
1, 23, Total = 1419
1, 24, Total = 1395
1, 25, Total = 1511
1, 26, Total = 1504
1, 27, Total = 1509
1, 28, Total = 1550
1, 29, Total = 1571
1, 30, Total = 1508
1, 31, Total = 1559
1, 32, Total = 1640
1, 33, Total = 1751
1, 34, Total = 1793
1, 35, Total = 1757
1, 36, Total = 1739
1, 37, Total = 1739
1, 38, Total = 1757
1, 39, Total = 1793
1, 40, Total = 1886
1, 41, Total = 1919
1, 42, Total = 1927
1, 43, Total = 1949
1, 44, Total = 1985
1, 45, Total = 2035
1, 46, Total = 2099
1, 47, Total = 2177
1, 48, Total = 2269
1, 49, Total = 2375
1, 50, Total = 2495
1, 51, Total = 2479
1, 52, Total = 2420
1, 53, Total = 2367
1, 54, Total = 2320
1, 55, Total = 2279
1, 56, Total = 2244
1, 57, Total = 2215
1, 58, Total = 2192
1, 59, Total = 2175
1, 60, Total = 2223
1, 61, Total = 2339
1, 62, Total = 2465
1, 63, Total = 2601
1, 64, Total = 2747
1, 65, Total = 2903
1, 66, Total = 3069
1, 67, Total = 3245
1, 68, Total = 3230
1, 69, Total = 3219
1, 70, Total = 3212
1, 71, Total = 3209
1, 72, Total = 3210
1, 73, Total = 3215
1, 74, Total = 3224
1, 75, Total = 3237
1, 76, Total = 3254
1, 77, Total = 3275
1, 78, Total = 3300
1, 79, Total = 3329
1, 80, Total = 3362
1, 81, Total = 3399
1, 82, Total = 3440
1, 83, Total = 3485
1, 84, Total = 3534
1, 85, Total = 3587
1, 86, Total = 3644
1, 87, Total = 3705
1, 88, Total = 3770
1, 89, Total = 3839
1, 90, Total = 3912
1, 91, Total = 3989
1, 92, Total = 4070
1, 93, Total = 4155
1, 94, Total = 4244
1, 95, Total = 4337
1, 96, Total = 4434
1, 97, Total = 4535
1, 98, Total = 4640
1, 99, Total = 4749
1, 100, Total = 4862
1, 101, Total = 4879
1, 102, Total = 4797
1, 103, Total = 4715
1, 104, Total = 4633
1, 105, Total = 4551
1, 106, Total = 4469
1, 107, Total = 4387
1, 108, Total = 4305
1, 109, Total = 4223
1, 110, Total = 4141
1, 111, Total = 4059
1, 112, Total = 3977
1, 113, Total = 3895
1, 114, Total = 3813
1, 115, Total = 3731
1, 116, Total = 3649
1, 117, Total = 3567
1, 118, Total = 3485
1, 119, Total = 3403
2, 3, Total = 4387
2, 5, Total = 2713
2, 7, Total = 2042
2, 9, Total = 1706
2, 11, Total = 1522
2, 13, Total = 1419
2, 15, Total = 1359
2, 17, Total = 1342
2, 19, Total = 1346
2, 21, Total = 1362
2, 23, Total = 1384
2, 25, Total = 1445
2, 27, Total = 1477
2, 29, Total = 1531
2, 31, Total = 1537
2, 33, Total = 1657
2, 35, Total = 1705
2, 37, Total = 1727
2, 39, Total = 1785
2, 41, Total = 1879
2, 43, Total = 1927
2, 45, Total = 2003
2, 47, Total = 2107
2, 49, Total = 2239
2, 51, Total = 2350
2, 53, Total = 2332
2, 55, Total = 2326
2, 57, Total = 2332
2, 59, Total = 2350
2, 61, Total = 2439
2, 63, Total = 2605
2, 65, Total = 2791
2, 67, Total = 2997
2, 69, Total = 3022
2, 71, Total = 3055
2, 73, Total = 3096
2, 75, Total = 3145
2, 77, Total = 3202
2, 79, Total = 3267
2, 81, Total = 3340
2, 83, Total = 3421
2, 85, Total = 3510
2, 87, Total = 3607
2, 89, Total = 3712
2, 91, Total = 3825
2, 93, Total = 3946
2, 95, Total = 4075
2, 97, Total = 4212
2, 99, Total = 4357
2, 101, Total = 4510
2, 103, Total = 4469
2, 105, Total = 4428
2, 107, Total = 4387
2, 109, Total = 4346
2, 111, Total = 4305
2, 113, Total = 4264
2, 115, Total = 4223
2, 117, Total = 4182
2, 119, Total = 4141
3, 4, Total = 3300
3, 5, Total = 2681
3, 7, Total = 2009
3, 8, Total = 1813
3, 10, Total = 1562
3, 11, Total = 1485
3, 13, Total = 1383
3, 14, Total = 1353
3, 16, Total = 1322
3, 17, Total = 1303
3, 19, Total = 1309
3, 20, Total = 1311
3, 22, Total = 1345
3, 23, Total = 1353
3, 25, Total = 1397
3, 26, Total = 1408
3, 28, Total = 1468
3, 29, Total = 1491
3, 31, Total = 1513
3, 32, Total = 1566
3, 34, Total = 1642
3, 35, Total = 1661
3, 37, Total = 1685
3, 38, Total = 1723
3, 40, Total = 1782
3, 41, Total = 1839
3, 43, Total = 1893
3, 44, Total = 1927
3, 46, Total = 2009
3, 47, Total = 2057
3, 49, Total = 2167
3, 50, Total = 2229
3, 52, Total = 2269
3, 53, Total = 2293
3, 55, Total = 2297
3, 56, Total = 2328
3, 58, Total = 2343
3, 59, Total = 2381
4, 5, Total = 2648
4, 7, Total = 1973
4, 9, Total = 1636
4, 11, Total = 1448
4, 13, Total = 1346
4, 15, Total = 1287
4, 17, Total = 1271
4, 19, Total = 1271
4, 21, Total = 1285
4, 23, Total = 1308
4, 25, Total = 1343
4, 27, Total = 1401
4, 29, Total = 1451
4, 31, Total = 1481
4, 33, Total = 1566
4, 35, Total = 1619
4, 37, Total = 1661
4, 39, Total = 1721
5, 6, Total = 2214
5, 7, Total = 1939
5, 8, Total = 1741
5, 9, Total = 1599
5, 11, Total = 1411
5, 12, Total = 1353
5, 13, Total = 1307
5, 14, Total = 1282
5, 16, Total = 1243
5, 17, Total = 1231
5, 18, Total = 1230
5, 19, Total = 1227
5, 21, Total = 1243
5, 22, Total = 1266
5, 23, Total = 1273
5, 24, Total = 1282
5, 26, Total = 1339
5, 27, Total = 1353
5, 28, Total = 1383
5, 29, Total = 1411
6, 7, Total = 1904
6, 11, Total = 1374
6, 13, Total = 1271
6, 17, Total = 1188
6, 19, Total = 1194
6, 23, Total = 1242
7, 8, Total = 1670
7, 9, Total = 1527
7, 10, Total = 1418
7, 11, Total = 1337
7, 12, Total = 1277
7, 13, Total = 1235
7, 15, Total = 1177
7, 16, Total = 1160
7, 17, Total = 1157
7, 18, Total = 1155
7, 19, Total = 1157
7, 20, Total = 1159
8, 9, Total = 1490
8, 11, Total = 1300
8, 13, Total = 1198
8, 15, Total = 1137
8, 17, Total = 1123
9, 10, Total = 1344
9, 11, Total = 1263
9, 13, Total = 1159
9, 14, Total = 1121
10, 11, Total = 1226
10, 13, Total = 1121
11, 12, Total = 1128
Last edited by: teliot on Feb 27, 2021
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ThatDonGuy
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February 27th, 2021 at 7:12:35 AM permalink
Quote: teliot

On the topic of Golfing, let me create a different problem that is based on efficient stamp totals.

Suppose the Post Office issues two values of stamps X cents and Y cents. What are the most efficient values for these two stamps X & Y so that we can make each postage from $1.20 to $2.00, using the least number of stamps overall?

Yeah, "Easy" Math Puzzles. :)


Because I have nothing better to do at 7 AM on a Saturday morning...


(13, 11) uses 1072 stamps:
Total1311Total1311Total1311Total1311
1205514048160311180131
1210111411011619418187
1226414257162410182140
12311014311016310318396
124731446616459184412
12529145112165112185105
126821467516668186511
12738147211167121187114
128911488416877188610
12947149310169130189123
130100150931708619079
1315615149171312191132
1320121521021729519288
1336515358173411193141
13411115411117410419497
1357415567175510195150
136210156120176113196106
137831577617769197512
13839158212178122198115
139921598517978199611
200124


teliot
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February 27th, 2021 at 7:30:55 AM permalink
Quote: ThatDonGuy

Because I have nothing better to do at 7 AM on a Saturday morning...


(13, 11) uses 1072 stamps:
Total1311Total1311Total1311Total1311
1205514048160311180131
1210111411011619418187
1226414257162410182140
12311014311016310318396
124731446616459184412
12529145112165112185105
126821467516668186511
12738147211167121187114
128911488416877188610
12947149310169130189123
130100150931708619079
1315615149171312191132
1320121521021729519288
1336515358173411193141
13411115411117410419497
1357415567175510195150
136210156120176113196106
137831577617769197512
13839158212178122198115
139921598517978199611
200124


Ha! I was writing c code at 5:30 this morning.

Correct, of course.
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February 27th, 2021 at 7:45:24 AM permalink
I'm sure we're all wondering if there is at least a short cut to the right answer to such problems or is brute force the only way? I suspect in general the coinages will be semi-prime to each other, but I can't put into words why.
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teliot
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February 27th, 2021 at 8:05:12 AM permalink
Quote: Wizard

I'm sure we're all wondering if there is at least a short cut to the right answer to such problems or is brute force the only way? I suspect in general the coinages will be semi-prime to each other, but I can't put into words why.

Yes, the coinages must be relatively prime. If d is a divisor of a & b, then d is also a divisor of ax + by for all x, y. Ergo, you can only make coinages that are divisible by gcd(a,b).

As for your interesting question of getting a short cut, that's why I gave the example of $1.19. The answer doesn't fit the most obvious short cut, namely to minimize the difference |a-b| while maximizing the product (a-1)*(b-1) <= N (where N is the smallest coinage that needs to be made).
Last edited by: teliot on Feb 27, 2021
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charliepatrick
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February 27th, 2021 at 8:52:25 AM permalink
This is only an observation but when the difference between two values was 2, one had to devise two sets of numbers.

For instance if you were using 5 and 7, 25 would be 5*5 and 0*7. Reducing the number of 5's by 1 and adding a 7 instead, would add 2 (27=4*5+1*7 etc.). So this creates 25 27 29 31 33 35. 28 is 4*7 so 26 is 3*7+1*5. 30 is 6*5 which starts the next series up to 42; 7*5 starts 35....

Presumably this process continues all the way up. So you might be able to determine how this goes based on the pattern and end conditions.
Gialmere
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March 1st, 2021 at 8:22:24 AM permalink
Here's an easy Monday puzzle...

Have you tried 22 tonight? I said 22.
unJon
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March 1st, 2021 at 8:23:53 AM permalink
Quote: Gialmere

Here's an easy Monday puzzle...



Did it quick in my head but looks like:

153
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teliot
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March 1st, 2021 at 8:46:15 AM permalink
Quote: Gialmere

Here's an easy Monday puzzle...

153 -- which, coincidentally, is the smallest integer > 1 that is the sum of the cubes of its digits. 153 = 1^3 + 5^3 + 3^3. There are three more positive integers > 1 with this property. Bonus easy Monday question -- What are they?

Bonus bonus question -- how many fish did Simon Peter catch (New Testament)?
Last edited by: teliot on Mar 1, 2021
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ThatDonGuy
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March 1st, 2021 at 9:11:30 AM permalink

By 891, one of 1, 8, 9 is in the code, but by 849, neither 8 nor 9 are in the code, so 1 is in the code, but not the third digit
By 317, 1 is not the second digit; therefore, 1 is the first digit
By 793, either 9 is the second digit or 3 is the third digit, but by 849, 9 is not in the code, so 3 is the third digit
By 725, either 7 or 5 is the second digit, but by 793, since 3 is in the code, 7 is not, so 5 is the second digit
The code is 153

unJon
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March 1st, 2021 at 9:23:12 AM permalink
Quote: ThatDonGuy


By 891, one of 1, 8, 9 is in the code, but by 849, neither 8 nor 9 are in the code, so 1 is in the code, but not the third digit
By 317, 1 is not the second digit; therefore, 1 is the first digit
By 793, either 9 is the second digit or 3 is the third digit, but by 849, 9 is not in the code, so 3 is the third digit
By 725, either 7 or 5 is the second digit, but by 793, since 3 is in the code, 7 is not, so 5 is the second digit
The code is 153



You can solve also without reference to the 849 box.

ETA:

Comparing 725 and 793 box you can eliminate 7. From 317 you then know 3 and 1 are correct but not in those positions. Looking at 793 you can put 3 in last position. Then looking at 891 leaves the 1 in first position. Finally looking at 725 you see that 5 has to be the middle digit.
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gordonm888
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March 1st, 2021 at 9:49:18 AM permalink
Quote: teliot

153 -- which, coincidentally, is the smallest integer > 1 that is the sum of the cubes of its digits. 153 = 1^3 + 5^3 + 3^3. There are three more positive integers > 1 with this property. Bonus easy Monday question -- What are they?

Bonus bonus question -- how many fish did Simon Peter catch (New Testament)?







370
371
407

At first, I thought the question was "prime numbers that are the sum of their digits," and I couldn't find any.

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teliot
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March 1st, 2021 at 11:34:07 AM permalink

0
1
153
370
371
407

Ok, here is a really easy follow-up question. The list in the spoiler gives all six solutions to "sum of the cubes of the digits equals the number."

My follow-up question is to find all integers that minimize the difference when it is greater than 0, that is:

minimize |(sum of cube of digits of number) - number| > 0

There are six solutions when this difference is equal to 0. I found 12 solutions to the next smallest difference.
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charliepatrick
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March 1st, 2021 at 12:55:26 PM permalink
Quote: teliot

...I found 12 solutions to the next smallest difference.

I could only find 11 - so still looking!
N: 12 C: 9 diff: 3
N: 30 C: 27 diff: 3
N: 31 C: 28 diff: 3
N: 32 C: 35 diff: 3
N: 255 C: 258 diff: 3
N: 365 C: 368 diff: 3
N: 437 C: 434 diff: 3
N: 474 C: 471 diff: 3
N: 747 C: 750 diff: 3
N: 856 C: 853 diff: 3
N: 1799 C: 1802 diff: 3
teliot
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March 1st, 2021 at 1:03:40 PM permalink
Quote: charliepatrick

I could only find 11 - so still looking!

N: 12 C: 9 diff: 3
N: 30 C: 27 diff: 3
N: 31 C: 28 diff: 3
N: 32 C: 35 diff: 3
N: 255 C: 258 diff: 3
N: 365 C: 368 diff: 3
N: 437 C: 434 diff: 3
N: 474 C: 471 diff: 3
N: 747 C: 750 diff: 3
N: 856 C: 853 diff: 3
N: 1799 C: 1802 diff: 3

I confess to miscounting! You got it. Here are the solutions with the difference <= 10:


1, 0
153, 0
370, 0
371, 0
407, 0
12, 3
30, 3
31, 3
32, 3
255, 3
365, 3
437, 3
474, 3
747, 3
856, 3
1799, 3
22, 6
226, 6
372, 6
1079, 6
10, 9
11, 9
125, 9
216, 9
417, 9
566, 9
675, 9
766, 9
872, 9
873, 9
962, 9
963, 9
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teliot
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March 1st, 2021 at 1:05:44 PM permalink
Okay, which leads me to a cute little theorem-ette that is on the slightly non-trivial side (but not that non-trivial), related to the 153 question.

Let N > 0 be any integer. Let S be the sum of the cubes of its digits. Show that the difference (N-S) is always divisible by 3.

For example, N = 1263. S = 1^3 + 2^3 + 6^3 + 3^3 = 1 + 8 + 216 + 27 = 252. N - S = 1011= 3*337.
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charliepatrick
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March 1st, 2021 at 1:27:42 PM permalink
We can split any number into its constituent digits and consider each one in turn. So a number such as 543 look at the difference between 500 and 53, 40 and 43, 3 and 33. If each of these is divisible by 3, then the total will.

So the approach is to show it for one digit numbers, then two digit numbers, etc.

Consider 0<N<10. N3-N = N * (N2-1) = N * (N-1) * (N+1). One these must be a multiple of 3.

Now consider a number <100 which is 10M+N. Then we would like M3-10M to be a multiple of 3. This is the same as (M3-M)-9*M. The former (M3-M) is divisible by 3 (as per above) and 9*M is also divisible by 3. Hence the difference is divisible by 3. This proves the case for two digit numbers.

Similar logic will apply to any other digit in the number since it will split into (L3-L)-999....999*L.

Hence it applies to all integers.
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