# Ciphers and Puzzles from Competitors

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• #49447
Keymaster

KA-9 Hint One
I presume you’ll be struggling with KA-9, so here is a hint – you may find a calculator handy!

#49452
Inactive

NT5- 87*97*94*89=70600674 ✓

@Prabhaa
NT4 23514624000 ✓
NT5 70600674 ✓
NT6 1297461358 x

#49453
Inactive

KA-9 = goes

Do the calcs on calculator and then read upside-down.

#49456
Keymaster

The KA Answers – KA-9 – Part 1
The answer is ‘goes’. My method of doing this is shown below! (Note: Skip to the bottom of this post for more problems relating to this!)

Section 1 – How to Find Out the Method

Here are the solved sums:
1879*3=legs
122*41=zoos
20*18000+1780=oblige
70038+2^4*3*5^2=bezil
(2*7+30*40+12000-20*100)/2=logs

First work out the answers to all the normal expressions:
1879*3=5637
122*41=5002
20*18000+1780=361780
70038+2^4*3*5^2=71238
(2*7+30*40+12000-20*100)/2=5607

So:
5637=legs
5002=zoos
361780=oblige
71238=bezil
5607=logs

Putting it this way really makes a difference. The first thing you may notice is the fact that both ‘logs’ and ‘legs’ are in there, and only one digit is changed – that is, the tens digit. So you would expect ‘e’ to be 3 and ‘o’ to be 0. But one is in the hundreds column and the other is in the tens! Maybe it wasn’t just a simple substitution in this step after all…

However, we can try solving this problem. Both ‘e’ and ‘o’ are in the word ‘oblige’. So we get something along these lines: ‘361780=0blig3’. This makes it clear that the numbers have to be reversed first before the substitution can be made. So:
0 is ‘o’
1 is ‘i’
2 is ‘z’
3 is ‘e’
5 is ‘s’
6 is ‘g’
7 is ‘l’
8 is ‘b’
(Note: the numbers 4 and 9 aren’t here simply because they didn’t feature in any of the answers to the sums. For a couple of problems relating to this, see the bottom of this post!)

Section 2 – Solving the Conundrum

So now we can solve the final sum! Solve it normally first:
2*(2500+3*51)=5306

Then reverse it:
5306 -> 6035

And finally, use the substitution above:
6 -> g
0 -> o
3 -> e
5 -> s

Hey presto! We have the word ‘goes’ as our answer!

Section 3 – How to Make Your Own Sum

In case you want to make your own puzzle and confuse your friends:

1. Find a word which only contains the letters ‘b’, ‘e’, ‘g’, ‘i’, ‘l’, ‘o’, ‘s’ and ‘z’. Make sure it doesn’t end in ‘o’ (for reasons you’re about to discover). (For example: the word ‘lose’.)
2. Use the substitution above. (For example: l -> 7; o -> 0; s -> 5; e -> 3.)
3. Reverse the number. (For example: 7053 -> 3507.)
4. Make a sum that has this number as its answer. Numbers can’t start with 0, so your word (Step 1) can’t end with ‘o’! (For example: 3507=501*7.)
5. Give your friend (or anyone else) the sum and watch them suffer! (The sum following the examples is 501*7=lose.)

Section 4 – Extra Conundrums

And now a few problems for you to ponder over:
1. The number 4 wasn’t mentioned in any of the answers to the sums. It actually represents a letter on its own. Can you find it?
2. The number 9 wasn’t mentioned in any of the answers to the sums, either. It actually represents the same letter as one of the other numbers. Which number?
3. You probably need to solve this problem to work out the above two! The substitution of the letters and numbers actually follows a pattern. What is it? (Hint One will come in handy for both this puzzle and the next one!)
4. There is a quicker way you can do this problem. Rather than reversing the number and using the substitution, there is a way you can skip those two steps and do it in one. What is it? (Note: this really only works for decrypting!)
The answers to these problems will be released in Part 2!

#49504
Inactive

Is anyone (still) trying to find the TLW MISSING NUMBERS? How many have you got so far?

TLW NUMBER THEORY HINTS here are some hints that may help…
=======================
NT6 Hint
========
This is going to be a big number – the last thing you want to do is print it out!
Better use the mod function wisely then.

NT7 Hint
========
What we are trying to find here is the sum of all the digits in a number
which gives the highest total for a^b where 1<= a <100 and 1<= b <100
(both a and b range from 1 to 99)

Example using two of the possible candidates:
5^90 = 807793566946316088741610050849573099185363389551639556884765625
5^91 =4038967834731580443708050254247865495926816947758197784423828125
when we add up the digits in 5^90 we get the sum 316
when we add up the digits in 5^91 we get the sum 311
so although 5^91 is greater than 5^90, the digit sum of 5^90 is the greater
and the answer would be 316.

NT8 Hint
========
Like NT6 but even more huge a number – what to do? Maybe use two mods?

#49538
Keymaster

TLW Missing Numbers
I’m struggling on #2, #4, #7 and #9!

#49540
Keymaster

I suppose @The-letter-wriggler nearly gave the answers away to these, and no-one seems to be taking any interest, so…

The KA Answers – KA-9 – Part 2
Here are the solutions to those extra questions!

1. 4 represents the letter ‘h’.
2. 9 represents the same letter as 6 (or the letter ‘g’).
3. Put one of the digits into a calculator and turn it upside-down. The result is (roughly in some cases) the letter it represents!
4. Similarly to the above problem, put the whole number into a calculator and turn it upside-down. The result is the word the number represents!

Section 2 – The Updated Substitution Chart

Indeed, here is the substitution chart (from Part 1, Section 1) with the numbers 4 and 9 added:
0 is ‘o’
1 is ‘i’
2 is ‘z’
3 is ‘e’
4 is ‘h’
5 is ‘s’
6 and 9 are ‘g’
7 is ‘l’
8 is ‘b’
(Although you can choose 9 for ‘g’, I prefer to use 6.)

Section 3 – The Updated Decryption Process

Basically Part 1, Section 2 condensed down!

1. Solve the problem normally. (For example: 2*(2500+3*51) is 5306.)
2. Put this number into your calculator and turn it upside-down! (For example: 5306 -> goes.)
3. Relax in the comfort of your home, knowing that you have cracked a very difficult puzzle!

Section 4 – The Updated Encryption Process

Basically Part 1, Section 3 varied slightly!

1. Find a word which only contains the letters ‘b’, ‘e’, ‘g’, ‘h’, ‘i’, ‘l’, ‘o’, ‘s’ and ‘z’. Make sure it doesn’t end in ‘o’ (for reasons you’re about to discover). (For example: the word ‘gosh’.)
2. Use the substitution above. (For example: g -> 6 or 9 (I choose 9); o -> 0; s -> 5; h -> 4.)
3. Reverse the number. (For example: 9054 -> 4509.)
4. Make a sum that has this number as its answer. Numbers can’t start with 0, so your word (Step 1) can’t end with ‘o’! (For example: 4509=501*9.)
5. Give your friend (or anyone else) the sum and watch them suffer! (The sum following the examples is 501*9=gosh.)

#49543
Inactive

TLW Missing Numbers
Providing your answers are correct that means you’ve done seven of them.
That really is good going so far, you are in with a chance of being 1st to post them all.

Those other 4 are not anymore the harder than the ones you have done, though when I first
encountered them it took me quite some time to solve them too, #7 being the tricky one.

#49545
Keymaster

NT8 – 8739992577. Key to working this out: knowledge that the last 10 digits in the sequence ‘2^n’ repeats every 7812500 terms, which is just 17957 less than the power in the expression (i.e. 7830457)!

#49552
Inactive

@Kford-academy Re: NT8 – 8739992577 correct

(You do NOT have to answer to the following friendly remarks if you do not want to)
Your notes tell me nothing as to how you actually worked it out!
No mention of the 28433* part of the math.
I’m intrigued, just how did you work it out I wonder?

I used a simple PARI-gp math formula that also works with python.

BTW Only the last 9 digits repeat every 7812500 terms thus:
2^7812500 = 1787109376 (last 10 digits)
7812500+7812500 = 15625000 (next term)
2^1562500 = 5787109376 (last 10 digits)
15625000+7812500 = 23437500 (next term)
2^23437500 = 1787109376 (last 10 digits)
etc.

#49568
Keymaster

The KA Answers – KA-8 (The Countdown Conundrum) – Part 2
Just a couple of notes first:
1. Can someone please continue this sequence during the holidays??? (I might not have time!)
2. When I said ‘you can make every number from 100 to 999’ in Part 1, I only assumed it! I at least know MOST of the numbers from 100 to 999 can be made, but I don’t know if ALL of them can be made!
150=10*(6+9)
151=10*(7+8)+6-5
152=10*(6+9)+7-5
153=10*(6+9)+8-5
154=10*(7+8)+9-5
155=10*(7+8)+5
156=10*(7+8)+6
157=10*(6+9)+7
158=10*(6+9)+8
159=10*(7+8)+9
160=10*(7+9)
161=10*(7+8)+6+5
162=10*(6+9)+7+5
163=10*(6+9)+8+5
164=10*(7+8)+9+5
165=10*(7+8)+9+6
166=10*8*(7-5)+6
167=10*8*(6+5-9)+7
168=10*(8+9)-7+5
169=10*(8+9)-6+5
170=10*(8+9)
171=10*(8+9)+6-5
172=10*(8+9)+7-5
173=7*(6+8+10)+5
174=6*(5+7+8+9)
175=10*(8+9)+5
176=10*(8+9)+6
177=10*(8+9)+7
178=10*(8+9)+7+6-5
179=9*(6+7+8)-10
180=10*9*(8-6)
181=9*(5+6+8)+10
182=7*(8*(9-5)-6)
183=9*(7*(8-5))+6
184=9*(6+7+8)-5
185=5*(10*(9-6)+7)
186=9*10*(7-5)+6
187=9*10*(8-6)+7
188=9*10*(7-5)+8
189=9*7*(8-5)
190=5*(10*(9-6)+8)
191=5*6*7-9-10
192=5*6*7-8-10
193=5*6*7-8-9
194=5*8*(10/(9-7))-6
195=5*(6*(7+8-10)+9)
196=10*(7+8+9-5)+6
197=10*(5*(8-6)+9)+7
198=6*(10*(9-5)-4)
199=9*7*(8-5)+10

The Ring
My solutions so far:
6-3-3+2-1=1, 63/3/21=1, 1^6332=1, 1^63*3-2=1, 1+6+3-3^2=1
1^2336=1, 1^2+3+3-6=1, 1^2-3-3+6=1, 12/3+3-6=1, 12*3/36=1
2+3+3-6-1=1, 2-3-3+6-1=1, 1+1+6-3-3=2, 1+1-6+3+3=2, 1*1-6/3+3=2
1/1-6/3+3=2, 1^1-6/3+3=2, 1^16+3/3=2, 1+1^633=2, 3+3-6+1+1=2
3-2+1+1^6=3, 2-1-1+6-3=3, 2-1*1+6/3=3, 2-1/1+6/3=3, 2-1^1+6/3=3
2+1^163=3, 2+1*1^63=3, 2+1/1^63=3, 2+1^1^63=3, 2*1+1^63=3
2/1+1^63=3, 2^1+1^63=3, 2*1-1+6/3=3, 2/1-1+6/3=3, 2^1-1+6/3=3
6-1-1+2-3=3, 6+1+1-2-3=3, 6*1*1^2-3=3, 6*1^1^2-3=3, 6+1-12/3=3
6^1*1^2-3=3, 6^1^1^2-3=3, 36*1/12=3, 36/1/12=3, 36^1/12=3
3+3+2-1-1=6, 3+3-2+1+1=6, 33*2/11=6, 1*12-3-3=6, 1*1*2*3+3=6
1*1^2*3+3=6, 1^1^2*3+3=6, 1^1*2*3+3=6, 6-3-3+2=1+1, 2-3-3+6=1+1
2+3+3-6=1+1, 6+3-3^2=1-1, 3*3-6*1=1+2, 3*3-6/1=1+2, 3*3-6^1=1+2
3+3-6-1=1-2, 3*3-6-1=1*2, 3*3/6-1=1/2, 3+3-6+1=1^2, 3+3+6*1=12
3+3+6/1=12, 3+3+6^1=12, 3*3-2*1=1+6, 3*3-2/1=1+6, 3*3-2^1=1+6
3+3+2-1=1+6, 3*3-2-1=1*6, 3^3-21=1*6, 3-3+2-1=1^6, 1^63*3=2+1
1+6-3-3=2-1, 1-6+3+3=2-1, 1^633=2-1, 1^6+3-3=2-1, 1^6-3+3=2-1
1^6*3/3=2-1, 1^6/3*3=2-1, 1-6/3+3=2*1, 1-6/3+3=2/1, 1-6/3+3=2^1
3-6+1+1=2-3, 3+6-1*1=2^3, 3+6-1/1=2^3, 3+6-1^1=2^3, 1+1+6-3=3+2
1^163=3-2, 1*1^63=3-2, 1/1^63=3-2, 1^1^63=3-2, 1+1/6*3=3/2
1*1*6+3=3^2, 1/1*6+3=3^2, 1^1*6+3=3^2, 1-1+6+3=3^2, 2+1+6-6=3*3
2+1-6+6=3*3, 21-6-6=3*3, 2-1^66=3/3, 21+6+6=33, 6+6-12=3-3
6-6*1^2=3-3, 6+6-1-2=3*3, 6/6*1^2=3/3, 6-6-1+2=3/3, 6/6^1^2=3/3
6-6+1^2=3/3, 66*1/2=33, 66/1/2=33, 66^1/2=33, 1*1+2^3=3+6
1/1+2^3=3+6, 1^1+2^3=3+6, 1*12-3=3+6, 1+1-2-3=3-6, 1-1^2-3=3-6
1*12*3=36, 12/3+3=6+1, 1-2+3+3=6-1, 1+2^3-3=6*1

#49604
Inactive

TLW ANSWERS TO MY POSTS – PART 2
================================

TLW High Score Self Made Crosswords
===================================
For the BLANK GRID only

I THINK ANYONE WILL BE HARD PUSHED TO BEAT THIS…

BLANK GRID – MY BEST SCORE 165 – USING 14 WORDS
===============================================
[P][A][X][_][J][_][P] 03+01+08+07+03 = 22
[_][X][_][Z][A][N][Y] 08+09+01+02+05 = 25
[F][E][Z][_][B][O][X] 04+01+09+03+01+08 = 26
[I][_][_][C][_][Z][_] 01+03+09 = 13
[Z][_][J][A][Z][Z][Y] 09+07+01+09+09+05 = 40
[Z][O][O][M][_][L][_] 09+01+01+04+02 = 17
[Y][_][Y][_][J][E][W] 05+05+07+01+04 = 22
=====================
TOTAL 22+25+26+13+40+17+22 = 165
================================
BUT IF YOU CAN, WELL…

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===============================================================================
TLW Snake Place Puzzles
=======================
Recreational: To fill a grid with snakes.

Convention: Reference is to the 1’s in a row, leftmost first.
A Snakes numbers follow the U up, D down, L left, R right path.
It is possible to have them in a different way to the ways given here.
==============================(#1)
1~~4
[4][3][X][4][1][2] R1 1>RDD
[1][2][1][3][X][3] R2 1>DRR, 1>LUL
[2][3][4][2][1][4] R3 1>LUU
[4][1][2][1][4][X] R4 1>RDD, 1>DDR
[3][X][3][2][3][2] R5 –
[2][1][4][3][4][1] R6 1>LUU, 1>ULU

==============================(#2)
1~~~5
[5][4][1][X][1][2][1] R1 1>DLUL, 1>RDLL, 1>DDLL
[5][3][2][5][4][3][2] R2 –
[4][X][1][5][5][4][3] R3 1>DDDD
[3][1][2][4][3][5][4] R4 1>DDDL
[2][2][3][1][2][X][3] R5 1>RULU
[1][3][4][1][2][1][2] R6 1>UUUU, 1>RDRR, 1>RUUL
[5][4][5][X][3][4][5] R7 –

==============================(#3)
1~~~~6
[5][6][3][4][2][3][4][5] R1 –
[4][1][2][5][1][6][X][6] R2 1>RURDD, 1>URRRD
[3][2][X][6][4][5][6][6] R3 –
[1][1][1][2][3][4][5][5] R4 1>DDDDR, 1>ULUUR, 1>RRURU
[2][5][4][1][2][3][3][4] R5 1>RRURU
[3][6][3][2][3][X][2][1] R6 1>LURUU
[4][X][2][1][4][5][6][6] R7 1>URDRR
[5][6][1][1][2][3][4][5] R8 1>UUULD, 1>RRRRU

==============================(#4)
1~~~~~7
[7][1][2][2][1][7][6][5][4] R1 1>RDLDDD, 1>LDDRRD
[6][4][3][3][X][7][6][1][3] R2 1>DDLUUL
[5][5][1][4][5][6][5][2][2] R3 1>DRRDRD
[4][6][2][3][4][7][4][3][1] R4 1>UUULLL
[3][7][X][7][5][6][X][2][1] R5 1>LDRDLD
[2][7][5][6][1][7][1][3][4] R6 1>DRDDLL, 1>DDDRRU
[1][6][4][3][2][3][2][6][5] R7 1>UUUUUU
[4][5][1][2][X][4][3][7][7] R8 1>RULURU
[3][2][1][7][6][5][4][5][6] R9 1>LLURUU

==============================(#5)
1~~4
[2][1][1][2][3][1][4][3][2][1] R1 1>LDD 1>RRD 1>DDL 1>LLL
[3][3][4][4][4][2][4][3][2][1] R2 1>LLL
[4][2][2][3][4][3][1][3][4][1] R3 1>DLL 1>DLD
[1][1][1][1][4][3][2][2][3][2] R4 1>DDR 1>UUR 1>URU 1>DLL
[2][4][3][2][2][1][4][1][4][1] R5 1>LDL 1>UUR 1>DLD
[3][4][4][4][3][1][3][2][3][2] R6 1>DDL
[1][2][3][2][1][2][1][1][4][4] R7 1>RRU 1>LDL 1>DDR 1>ULU
[3][4][4][3][4][3][2][1][2][3] R8 1>RRU
[2][3][2][1][1][1][3][4][1][4] R9 1>LLD 1>DLL 1>DRR 1>DRU
[1][4][4][3][2][2][3][4][2][3] R10 1>UUR

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===============================================================================
TLW A NUMBER RING
=================
Reminder:
Going clockwise or counterclockwise the ring of numbers
form a mathematical expression when certain empty cells
are filled with + – / * ^ = and an E for End.
If you leave a blank between digits then the digits
concatenate to form a multi-digit number.
All digits have to be used.

THE RING
[_][3][_][2]
[3][X][X][_]
[_][X][X][1]
[6][_][1][_]

The six that I gave as sample solutions are:
33*2/11=6, 1*6*3+3=21, 3+3-2+1+1=6, 3^3-21*1=6, 3*3-6-1*1=2, 6*11/2=33

A contestant, Kford-academy, found and gave the following 124 solutions.

My solutions so far:
6-3-3+2-1=1, 63/3/21=1, 1^6332=1, 1^63*3-2=1, 1+6+3-3^2=1
1^2336=1, 1^2+3+3-6=1, 1^2-3-3+6=1, 12/3+3-6=1, 12*3/36=1
2+3+3-6-1=1, 2-3-3+6-1=1, 1+1+6-3-3=2, 1+1-6+3+3=2, 1*1-6/3+3=2
1/1-6/3+3=2, 1^1-6/3+3=2, 1^16+3/3=2, 1+1^633=2, 3+3-6+1+1=2
3-2+1+1^6=3, 2-1-1+6-3=3, 2-1*1+6/3=3, 2-1/1+6/3=3, 2-1^1+6/3=3
2+1^163=3, 2+1*1^63=3, 2+1/1^63=3, 2+1^1^63=3, 2*1+1^63=3
2/1+1^63=3, 2^1+1^63=3, 2*1-1+6/3=3, 2/1-1+6/3=3, 2^1-1+6/3=3
6-1-1+2-3=3, 6+1+1-2-3=3, 6*1*1^2-3=3, 6*1^1^2-3=3, 6+1-12/3=3
6^1*1^2-3=3, 6^1^1^2-3=3, 36*1/12=3, 36/1/12=3, 36^1/12=3
3+3+2-1-1=6, 3+3-2+1+1=6, 33*2/11=6, 1*12-3-3=6, 1*1*2*3+3=6
1*1^2*3+3=6, 1^1^2*3+3=6, 1^1*2*3+3=6, 6-3-3+2=1+1, 2-3-3+6=1+1
2+3+3-6=1+1, 6+3-3^2=1-1, 3*3-6*1=1+2, 3*3-6/1=1+2, 3*3-6^1=1+2
3+3-6-1=1-2, 3*3-6-1=1*2, 3*3/6-1=1/2, 3+3-6+1=1^2, 3+3+6*1=12
3+3+6/1=12, 3+3+6^1=12, 3*3-2*1=1+6, 3*3-2/1=1+6, 3*3-2^1=1+6
3+3+2-1=1+6, 3*3-2-1=1*6, 3^3-21=1*6, 3-3+2-1=1^6, 1^63*3=2+1
1+6-3-3=2-1, 1-6+3+3=2-1, 1^633=2-1, 1^6+3-3=2-1, 1^6-3+3=2-1
1^6*3/3=2-1, 1^6/3*3=2-1, 1-6/3+3=2*1, 1-6/3+3=2/1, 1-6/3+3=2^1
3-6+1+1=2-3, 3+6-1*1=2^3, 3+6-1/1=2^3, 3+6-1^1=2^3, 1+1+6-3=3+2
1^163=3-2, 1*1^63=3-2, 1/1^63=3-2, 1^1^63=3-2, 1+1/6*3=3/2
1*1*6+3=3^2, 1/1*6+3=3^2, 1^1*6+3=3^2, 1-1+6+3=3^2, 2+1+6-6=3*3
2+1-6+6=3*3, 21-6-6=3*3, 2-1^66=3/3, 21+6+6=33, 6+6-12=3-3
6-6*1^2=3-3, 6+6-1-2=3*3, 6/6*1^2=3/3, 6-6-1+2=3/3, 6/6^1^2=3/3
6-6+1^2=3/3, 66*1/2=33, 66/1/2=33, 66^1/2=33, 1*1+2^3=3+6
1/1+2^3=3+6, 1^1+2^3=3+6, 1*12-3=3+6, 1+1-2-3=3-6, 1-1^2-3=3-6
1*12*3=36, 12/3+3=6+1, 1-2+3+3=6-1, 1+2^3-3=6*1

###############################################################################
===============================================================================
TLW A Number To Find
=====================

09==06
==06==
11==??

If we letter the square thus:
A=B
=E=
D=C

Then the 4 corner numbers relationship to the centre square number is:
((D*B)-C)/A = E
so
((11*6)-C)/9 = 6
rearrange
(D*B)-(A*E) = C
(11*6)-(9*6) = 12

###############################################################################
===============================================================================
TLW THE HUTCHINSON PROBLEM
==========================
2 3 3 2 1 4
1 2 3 1 2 3
3 2 1 1 3 3
1 1 3 2 3 4
2 2 4 3 4 2
3 4 3 3 2 X

The shortest path?
Here is one solution: 2D-3L-1L-3D-2L-3R-X (6 steps)

2 0 0 0 0 0
0 0 0 0 0 0
3 0 0 1 3 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 3 0 2 X

Open question, The longest path of jumps?
Open question, How many different unique paths?

###############################################################################
==========================================================
TLW A comPLEXmatrix
===================
Very difficult.
You were given X=4, P=3 and E=1

X P L E L
P E L E P
P L X E X
L L L E ?
E E X L L

One possible solution is
X=4
P=3
L=2
E=1

In each row, the number assigned to the rightmost letter is equal to the number assigned to the first letter minus the second, plus the third, minus the forth.
therefore, one solution for the missing letter is 1 = E.

4 3 2 1 2 [4-3+2-1=2]
3 1 2 1 3 [3-1+2-1=3]
3 2 4 1 4 [3-2+4-1=4]
2 2 2 1 1 [2-2+2-1=1]=E
1 1 4 2 2 [1-1+4-2=2]

Open question, are there other solutions?

###############################################################################
===============================================================================
TLW TRIPLET TRIP
================
A B C A C A
C A C B C B
B B A C A B
A C B B A C

Here is one possible answer using C-A-B

17 18 19 20 01 02
16 23 22 21 04 03
15 24 11 10 05 06
14 13 12 09 08 07

The only other is the reverse of this using B-A-C

###############################################################################
==================================================================== NT1 to NT3
TLW NUMBER THEORY
=================
The largest palindrome made from the product of two 3-digit numbers.
==========================
Answer 913 x 993 = 906609
==========================
The smallest positive number that is evenly divisible
by all of the numbers from 1 to 20.
=================
=================
The difference between the sum of the squares of the
first one hundred natural numbers and the square of the sum.
=================
=================

###############################################################################
==================================================================== NT4 to NT8
TLW NUMBER THEORY CONTINUED
===========================
Largest product in a series
The thirteen adjacent digits in the 1000-digit number that have
the greatest product.
What is the value of this product?
(With 987 possibles we find it starting at digit position 198:
5*5*7*6*6*8*9*6*6*4*8*9*5 = 23514624000)
==================
==================

The greatest product of four adjacent numbers in the
given 20 x 20 grid.
(There are 20*16+20*16+17*17+17*17 = 1218 possible quadlets)
==============================
==============================

Largest product in a grid extra info.

49*95*71*99 = 32719995 highest from all right sloping diagonals
66*91*88*97 = 51267216 highest from all columns
78*78*96*83 = 48477312 highest from all rows
89*94*97*87 = 70600674 highest from all left sloping diagonals (highest overall)

08 02 22 97 38 15 00 40 00 75 04 05 07 78 52 12 50 77 91 08
49 49 99 40 17 81 18 57 60 87 17 40 98 43 69 48 04 56 62 00
81[49]31 73 55 79 14 29 93 71 40 67 53 88 30 03 49 13 36 65
52 70[95]23 04 60 11 42 69 24 68 56 01 32 56 71 37 02 36 91
22 31 16[71]51 67 63 89 41 92 36 54 22 40 40 28 66 33 13 80
24 47 32 60[99]03 45 02 44 75 33 53 78 36 84 20 35 17 12 50
32 98 81 28 64 23 67 10 26 38 40 67 59 54 70[66]18 38 64 70
67 26 20 68 02 62 12 20 95 63 94 39 63 08 40[91]66 49 94 21
24 55 58 05 66 73 99 26 97 17[78 78 96 83]14[88]34 89 63 72
21 36 23 09 75 00 76 44 20 45 35 14 00 61 33[97]34 31 33 95
78 17 53 28 22 75 31 67 15 94 03 80 04 62 16 14 09 53 56 92
16 39 05 42 96 35 31 47 55 58 88 24 00 17 54 24 36 29 85 57
86 56 00 48 35 71[89]07 05 44 44 37 44 60 21 58 51 54 17 58
19 80 81 68 05[94]47 69 28 73 92 13 86 52 17 77 04 89 55 40
04 52 08 83[97]35 99 16 07 97 57 32 16 26 26 79 33 27 98 66
88 36 68[87]57 62 20 72 03 46 33 67 46 55 12 32 63 93 53 69
04 42 16 73 38 25 39 11 24 94 72 18 08 46 29 32 40 62 76 36
20 69 36 41 72 30 23 88 34 62 99 69 82 67 59 85 74 04 36 16
20 73 35 29 78 31 90 01 74 31 49 71 48 86 81 16 23 57 05 54
01 70 54 71 83 51 54 69 16 92 33 48 61 43 52 01 89 19 67 48

=============================================== HINTS GIVEN FOR NT6,NT7,NT8
Here are the hints that were given…
=====================================
NT6 Hint
========
This is going to be a big number – the last thing you want to do is print it out!
Better use the mod function wisely then.

NT7 Hint
========
What we are trying to find here is the sum of all the digits in a number
which gives the highest total for a^b where 1<= a <100 and 1<= b <100
(both a and b range from 1 to 99)

Example using two of the possible candidates:
5^90 = 807793566946316088741610050849573099185363389551639556884765625
5^91 =4038967834731580443708050254247865495926816947758197784423828125
when we add up the digits in 5^90 we get the sum 316
when we add up the digits in 5^91 we get the sum 311
so although 5^91 is greater than 5^90, the digit sum of 5^90 is the greater
and the answer would be 316.

NT8 Hint
========
Like NT6 but even more huge a number – what to do? Maybe use two mods?

Self powers
===========
Sum of all self powers to 1000 (1^1 to 1000^1000). Last 10 digits.
=================
=================

Powerful digit sum
==================
Considering natural numbers of the form, a^b, where a, b < 100,
what is the maximum digital sum?
==========
==========

Large Non-Mersenne Prime
========================
Non-Mersenne prime 28433×2^7830457+1 which contains 2,357,207 digits.

Find the last ten digits of this prime number.
=================
=================
================================================================

If you like this kind of math you will be pleased to know there are over 700 more
at http://www.projecteuler.net The Problems 3,5,6,8,11,48,56 and 97 are the ones I gave you.

================================================================
================================================================ WORKINGS

I use the free Maths Package PARI-gp for nearly all my Number Theory calculations.
I supply the scrips I wrote for the calculations.

You will see the output of these scripts if you copy them and then paste them at
http://pari.math.u-bordeaux.fr/gpwasm.html

====================================================== NT1 PARI-gp Script
\\The largest palindrome made from the product of two 3-digit numbers.
\\==========================
\\These scripts AVOID DUPLICATES like
\\101 x 121 &
\\121 x 101

\\The following finds and prints the HIGHEST palindrome…

b=0;for(i=100,999,for(j=i,999,s=i*j;n=digits(s);a=n;x=length(n);y=x;for(k=1,x,a[k]=n[y];y-=1);if(a==n,if(s>b,b=s;u=i;v=j))));print(“The Largest Palindrome Is: “u” x “v” = “b)

\\The following prints ALL the palindromes and the HIGHEST one with the 2 numbers…

b=0;for(i=100,999,for(j=i,999,s=i*j;n=digits(s);a=n;x=length(n);y=x;for(k=1,x,a[k]=n[y];y-=1);if(a==n,if(s>b,b=s;u=i;v=j);print(i” x “j” “n” = “a))));print(“The Largest Palindrome Is: “u” x “v” = “b)

====================================================== NT2 PARI-gp Script
\\The smallest positive number that is evenly divisible
\\by all of the numbers from 1 to 20.
\\=================
\\Smallest multiple is the LCM of the 20 numbers.
\\using the lcm function in gp…
a=lcm([1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]);
print(a,” is divisible by all the numbers 1 to 20″)
print(“Proof:”)
for(n=1,20,print(“232792560 divided by “n” = “232792560/n))

====================================================== NT3 PARI-gp Script
\\The difference between the sum of the squares of the
\\first one hundred natural numbers and the square of the sum.
\\=================
\\i.e. sum(n)^2-sum(n^2) where n = 1..100

\\By calculation:
(((100*101)/2)^2)-((2*100+1)*(100+1)*100/6) = 25164150

\\By program to allow for variable range:
range=100;zum=range*(range+1)/2;zumsq=(2*range+1)*(range+1)*range/6;print(zum^2-zumsq);

\\Program by iteration method:
p=q=0;for(n=1,100,p+=n^2;q+=n);print(q^2-p);

\\With print out:
p=q=0;for(n=1,100,p+=n^2;q+=n);print(“Square of Sum “q^2” minus Sum of Squares “p” is “q^2-p);
\\Square of Sum 25502500 minus Sum of Squares 338350 is 25164150

\\for 1000
range=1000;zum=range*(range+1)/2;zumsq=(2*range+1)*(range+1)*range/6;print(zum^2-zumsq);

====================================================== NT4 PARI-gp Script
\\Largest product in a series
\\======== START OF SCRIPT ========
d=7316717653133062491922511967442657474235534919493496983520312774506326239578318016984801869478851843858615607891129494954595017379583319528532088055111254069874715852386305071569329096329522744304355766896648950445244523161731856403098711121722383113622298934233803081353362766142828064444866452387493035890729629049156044077239071381051585930796086670172427121883998797908792274921901699720888093776657273330010533678812202354218097512545405947522435258490771167055601360483958644670632441572215539753697817977846174064955149290862569321978468622482839722413756570560574902614079729686524145351004748216637048440319989000889524345065854122758866688116427171479924442928230863465674813919123162824586178664583591245665294765456828489128831426076900422421902267105562632111110937054421750694165896040807198403850962455444362981230987879927244284909188845801561660979191338754992005240636899125607176060588611646710940507754100225698315520005593572972571636269561882670428252483600823257530420752963450;
\\set a to be a vector
a=digits(d);\
g=0;n=m=1;range=13;k=length(a)-range;\
for(j=1,k-1,m=a[j]*a[j+1]*a[j+2]*a[j+3]*a[j+4]*a[j+5]*a[j+6]*a[j+7]*a[j+8]*a[j+9]*a[j+10]*a[j+11]*a[j+12];if(g<m,g=m;n=j));\
print1(“Starting at digit: “,n,” “);for(x=n,n+12,print1(a[x]);if(x<n+12,print1(“*”)));print1(” = “,g)

\\======== END OF SCRIPT ========

====================================================== NT5 PARI-gp Scripts

\\==== FIND THE HIGHEST PRODUCT OF FOUR CONSECUTIVE NUMBERS IN A GRID ====

\\======== START OF SCRIPT ========
\\Set a[] to a single vector of the numbers
a=[08,02,22,97,38,15,00,40,00,75,04,05,07,78,52,12,50,77,91,08,49,49,99,40,17,81,18,57,60,87,17,40,98,43,69,48,04,56,62,00,81,49,31,73,55,79,14,29,93,71,40,67,53,88,30,03,49,13,36,65,52,70,95,23,04,60,11,42,69,24,68,56,01,32,56,71,37,02,36,91,22,31,16,71,51,67,63,89,41,92,36,54,22,40,40,28,66,33,13,80,24,47,32,60,99,03,45,02,44,75,33,53,78,36,84,20,35,17,12,50,32,98,81,28,64,23,67,10,26,38,40,67,59,54,70,66,18,38,64,70,67,26,20,68,02,62,12,20,95,63,94,39,63,08,40,91,66,49,94,21,24,55,58,05,66,73,99,26,97,17,78,78,96,83,14,88,34,89,63,72,21,36,23,09,75,00,76,44,20,45,35,14,00,61,33,97,34,31,33,95,78,17,53,28,22,75,31,67,15,94,03,80,04,62,16,14,09,53,56,92,16,39,05,42,96,35,31,47,55,58,88,24,00,17,54,24,36,29,85,57,86,56,00,48,35,71,89,07,05,44,44,37,44,60,21,58,51,54,17,58,19,80,81,68,05,94,47,69,28,73,92,13,86,52,17,77,04,89,55,40,04,52,08,83,97,35,99,16,07,97,57,32,16,26,26,79,33,27,98,66,88,36,68,87,57,62,20,72,03,46,33,67,46,55,12,32,63,93,53,69,04,42,16,73,38,25,39,11,24,94,72,18,08,46,29,32,40,62,76,36,20,69,36,41,72,30,23,88,34,62,99,69,82,67,59,85,74,04,36,16,20,73,35,29,78,31,90,01,74,31,49,71,48,86,81,16,23,57,05,54,01,70,54,71,83,51,54,69,16,92,33,48,61,43,52,01,89,19,67,48];
\\====================
\\By Rows
b=[0,0,0,0];\
zum=maxzum=row=0;\
for(n=1,400-4,zum=a[n]*a[n+1]*a[n+2]*a[n+3];\
if(zum>maxzum,maxzum=zum;b[1]=a[n];b[2]=a[n+1];b[3]=a[n+2];b[4]=a[n+3];row=maxzum));\
print(“Highest in a row “b” = “row)
\\====================
\\By Columns…
b=[0,0,0,0];\
zum=maxzum=col=0;n=1;\
until(n>400-61,zum=a[n]*a[n+20]*a[n+40]*a[n+60];\
if(zum>maxzum,maxzum=zum;b[1]=a[n];b[2]=a[n+20];b[3]=a[n+40];b[4]=a[n+60];col=maxzum);n+=1);\
print(“Highest in a column “b” = “col)
\\====================
\\By Diagonals Left-Right sloping
b=[0,0,0,0];\
zum=maxzum=dlr=0;p=17;n=1;\
until(n>300,if(n%p==0,n+=3);zum=a[n]*a[n+21]*a[n+42]*a[n+63];\
if(zum>maxzum,maxzum=zum;b[1]=a[n];b[2]=a[n+21];b[3]=a[n+42];b[4]=a[n+63];dlr=maxzum);n+=1);\
print(“Highest in a left-right sloping diagonal “b” = “dlr)
\\====================
\\By Diagonals Right-Left sloping
b=[0,0,0,0];\
zum=maxzum=drl=0;p=20;n=3;\
until(n>300,if(n%p==0,n+=3);zum=a[n]*a[n+19]*a[n+38]*a[n+57];\
if(zum>maxzum,maxzum=zum;b[1]=a[n];b[2]=a[n+19];b[3]=a[n+38];b[4]=a[n+57];drl=maxzum);n+=1);\
print(“Highest in a right-left sloping diagonal “b” = “drl)
\\====================
\\Confirm the largest product

\\======== END OF SCRIPT ========

====================================================== NT6 PARI-gp Script
\\Self powers
\\===========
\\Sum of all self powers to 1000 (1^1 to 1000^1000). Last 10 digits.
\\== Start ==
s=0;for(n=1,1000,s+=n^n);s%10000000000
\\== End ==

======================================== python script
s=0;
for n in range(1,1000):
#next 2 lines should have tab indents
[tab]s+=n**n;
[tab]s=s%10000000000;
print(s)
====================================================== NT7 PARI-gp Script
\\Powerful digit sum
\\==================
\\Considering natural numbers of the form, a^b, where a, b < 100,
\\what is the maximum digital sum?

\\== Start ==
zum=maxzum=0;for(a=1,99,for(b=1,99,zum=vecsum(digits(a^b));if(zum>maxzum,maxzum=zum;c=a;d=b)));print(c”^”d” = “maxzum)
\\== End ==

====================================================== NT8 PARI-gp Script
\\Large non-Mersenne prime
\\The last 10 digits of the prime 28433*2^7830457+1
\\== Start ==
(28433*(2^7830457%10000000000)+1)%10000000000
\\== End ==

======================================== python script:
a=(28433*(2**7830457%10000000000)+1)%10000000000
print(a)

###############################################################################
===============================================================================
TLW MISSING NUMBERS
===================
1] 909
2] 10
3] 28
4] 32
5] 102
6] 24
7] 2
8] 9
9] 43
10] 36
11] 360

HOW THEY WERE WORKED OUT.
=============================================================================== #1
909 ([9]90 > 90[9])
Rotated digits of the bottom row.
=============================================================================== #2
10 (7+9-6=10)
Vertical pattern applies to each column.
Add 1st row and 2nd row numbers and subtract 6 for 3rd row number.
=============================================================================== #3
28 (28/7=4)
Vertical pattern applies to consecutive pairs only.
Going down the two pairs divide 1st number by 7 to get 2nd number.
Or going up the two pairs multiply 1st number by 7 to get 2nd.
=============================================================================== #4
32 (12+5=17
Horizontal pattern applies to consecutive pairs only.
Reverse digits of 1st number and add 5 to get 2nd number.
=============================================================================== #5
102 (19+19=38,38+9=57)
Horizontal pattern applies to each row.
1st number is added to itself to get 2nd number, added again to get 3rd number.
=============================================================================== #6
24 (5*2=10,10-1=9)
Sarting at the right, double top number and subtract 1 to get the bottom number.
=============================================================================== #7
2 (6/1=6)
Horizontal pattern applies to consecutive pairs only.
2nd digit of 1st number is divided by 1st digit of 1st number to obtain the 2nd number.
=============================================================================== #8
9 (101+1=102;47+55=102)
Pattern applies to horizontal lines only.
Sum of outside numbers is equal to the sum of inside numbers.

=============================================================================== #9
43 (3*3=9-5=4)
Starting with smallest number,
multiply each number by three and
subtract five to get next number.
=============================================================================== #10
36 (4*9=36)
All numbers are multiplied by 9
=============================================================================== #11
360
Row1 (3*1=3;3*2=6;6*3=18)
Row2 [72*5=360;360*6=02160;2160*7=15120]

Horizontal pattern applies to each row (r1)[r2].
1st number multiplied by (1) [5] to get 2nd number,
2nd number multiplied by (2) [6] to get 3rd number,
3rd number multiplied by (3) [7] to get 4th number.

###############################################################################
===============================================================================

#49624
Keymaster

Some new combination-lock challenges. The first uses a mixed ciphertext alphabet, but
a straight plaintext alphabet. The second uses the same mixed alphabet for both.
The third uses two different mixed alphabets. Have fun.

[email protected]@TNTPGTQCCHIERKIOVRXETCGUSCYY
[email protected][email protected]@DNNJTM
@[email protected]@[email protected]@[email protected]NFHT
[email protected]@[email protected]
[email protected]@N[email protected][email protected]
[email protected]@[email protected]@[email protected]@[email protected]@VQKBUE
[email protected]@QJSPTMYHTCBESUFNFCOSQPHREQINNNQURNODMWRSWRFGRQOSRSIQC
[email protected]@[email protected]@OPRCESQRTNRC[email protected]
[email protected]@[email protected]@VOVKRUNOFINPTZCTRE
[email protected]AJDNXCTSCJNEUXOGNTQW
FFJ

[email protected]@[email protected]
[email protected][email protected]
[email protected]@[email protected]@[email protected]@@ODEUOJE
[email protected]@[email protected]@ACNAAY