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Crossnumber

cinelli
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Crossnumber

#166861

Postby cinelli » September 17th, 2018, 10:19 am

-------------------------------------------------
|A | |B | | |C | | |
| | | | | | | | |
-------------------------------------------------
| |XXXXX| |XXXXX|XXXXX| |XXXXX|XXXXX|
| |XXXXX| |XXXXX|XXXXX| |XXXXX|XXXXX|
-------------------------------------------------
|D | | |E |XXXXX|F | |XXXXX|
| | | | |XXXXX| | |XXXXX|
-------------------------------------------------
| |XXXXX|XXXXX| |XXXXX| |XXXXX|XXXXX|
| |XXXXX|XXXXX| |XXXXX| |XXXXX|XXXXX|
-------------------------------------------------
|G | | | |XXXXX|XXXXX|XXXXX|XXXXX|
| | | | |XXXXX|XXXXX|XXXXX|XXXXX|
-------------------------------------------------

Across
A = P!/Q
D = N^R*(N^S-1)-T
F = S*P-R
G = S!-N*W+1
Down
A = W^N
B = N*T
C = N^P+Q^N
E = W+T

Cinelli

Gengulphus
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Re: Crossnumber

#167043

Postby Gengulphus » September 17th, 2018, 10:55 pm

cinelli wrote:
-------------------------------------------------
|A | |B | | |C | | |
| | | | | | | | |
-------------------------------------------------
| |XXXXX| |XXXXX|XXXXX| |XXXXX|XXXXX|
| |XXXXX| |XXXXX|XXXXX| |XXXXX|XXXXX|
-------------------------------------------------
|D | | |E |XXXXX|F | |XXXXX|
| | | | |XXXXX| | |XXXXX|
-------------------------------------------------
| |XXXXX|XXXXX| |XXXXX| |XXXXX|XXXXX|
| |XXXXX|XXXXX| |XXXXX| |XXXXX|XXXXX|
-------------------------------------------------
|G | | | |XXXXX|XXXXX|XXXXX|XXXXX|
| | | | |XXXXX|XXXXX|XXXXX|XXXXX|
-------------------------------------------------

Across
A = P!/Q
D = N^R*(N^S-1)-T
F = S*P-R
G = S!-N*W+1
Down
A = W^N
B = N*T
C = N^P+Q^N
E = W+T

Spoiler...

I think I'll start by looking at N, T and W. We know that A_down = W^N is a 5-digit number, which tells us that W is neither 0 nor 1 and N is not 0. So A_down >= 2^N, which implies that N <= 16.

For each possible value of N from 1 to 16, the fact that B = N*T is a 3-digit number gives us bounds RoundUp(100/N) and RoundDown(999/N) on T. From those bounds on T, we can then use the fact that E = W+T is a 3-digit number to give us bounds Maximum(2,100-MaximumOfT) = Maximum(2,100-RoundDown(999/N)) and 999-MinimumOfT = 999-RoundUp(100/N) on W. That gives us minimum and maximum bounds of (Maximum(2,100-RoundDown(999/N)))^N and (999-RoundUp(100/N))^N on A_down. These are incompatible with A_down being a 5-digit number if N=1, since that makes the maximum bound be 899, or if N >= 11, since that makes the minimum bound >= 10^11. So 2 <= N <= 10.

For each of the remaining possible values of N, we can use the fact that A_down = W^N is a 5-digit number to determine possible values of W. They are:

N=2: W is in the range 100 to 316
N=3: W is in the range 22 to 46
N=4: W is in the range 10 to 17
N=5: W is in the range 7 to 9
N=6: W is 5 or 6
N=7: W is 4 or 5
N=8: W is 4
N=9: W is 3
N=10: W is 3

This tells us that 27 <= N*W <= 632. Then the fact that G = S!-N*W+1 is a 4-digit number and looking at the sequence of factorials 0! = 1, 1! = 1, 2! =2, 3! = 6, 4! = 24, 5! = 120, 6! = 720, 7! = 5040, 8! = 40320, ... tells us that S must be 7 and the last digit of A_down = W^N must be 4 if W*N > 5040+1-5000 = 41 or 5 if W*N <= 41. That allows us to eliminate a lot of the combinations of N and W, leaving us with the following possibilities:

N=2: W is in the range 100 to 316 and has last digit 2 or 8
N=3: W is 24, 34 or 44
N=6: W is 5
N=7: W is 5

For each of those values of N, the bounds RoundUp(100/N) and RoundDown(999/N) on T implied by B = N*T being a 3-digit number are:

N=2: 50 <= T <= 499
N=3: 34 <= T <= 333
N=6: 17 <= T <= 166
N=7: 15 <= T <= 142

Now look at the implications of D = N^R*(N^S-1)-T being a 4-digit number. We know that S=7, so for N=6, D = 6^R * (6^7-1) - T = 6^R * 279935 - T implies that D is hugely in excess of being a 4-digit number even if R=0 and T has its maximum possible value, and similarly for N=7 and D = 7^R * (7^7-1) - T = 6^R * 823542 - T. So neither N=6 nor N=7 is possible.

What about N=3? In that case, D = 3^R * (3^7-1) - T = 3^R * 2186 - T can be a 4-digit number if R=0, in which case the first digit of D is 1 if 187 <= T < 333 and 2 if 34 <= T <= 186, or if R=1, in which case the first digit of D is 6. But the first digit of D is also the middle digit of A_down and A_down is one of 24^3 = 13824, 34^3 = 39304 or 44^3 = 85184, so if N=3, we can deduce that R=0, W=44 and 187 <= T < 333. That tells us that G = 7! - 3*44 + 1 = 4909, so E = 44+T ends on 9, which implies that T ends on 5, and therefore that D = 3^0 * 2186 - T ends on 1. So E = 44+T starts with 1, but that's incompatible with 187 <= T < 333. So N=3 isn't possible, and we must have N=2, W in the range 100 to 316 and ending with 2 or 8, and 50 <= T <= 499.

We also have D = 2^R * (2^7-1) - T = 2^R * 127 - T. For T in the required range, that must be less than 4 digits long if R=0, 1, 2 or 3, and more than 4 digits long if R >= 7. For R=4, 5 or 6, it must be 4 digits long, with first digit (= middle digit of A_down = W^2, which allows me to somewhat tediously determine the possibilities for W in each case by squaring each of the 43 numbers between 100 and 316 and ending on 2 or 8):

1 if R=4 (and 50 <= T <= 499), with the possibilities for W being 142, 152 or 182
3 if R=5 and 65 <= T <= 499, with the possibilities for W being 128, 188 or 312
4 if R=5 and 50 <= T <= 64, with the possibilities for W being 102 or 132
7 if R=6 and 129 <= T <= 499, with there being no possibilities for W
8 if R=6 and 50 <= T <= 128, with the possibilities for W being 122, 192, 202, 232, 268, 298 or 308

If W ends on 2, then G = 7! - 2*W + 1 ends on 7. So E = W+T ends on 7 and so T ends on 5. So if R=4, D = 2032-T ends on 7; if R=5, D = 4064-T ends on 9, and if R=6, D = 8128-T ends on 3.

If W ends on 8, then G = 7! - 2*W + 1 ends on 5. So E = W+T ends on 5 and so T ends on 7. So if R=4, D = 2032-T ends on 5; if R=5, D = 4064-T ends on 7, and if R=6, D = 8128-T ends on 1.

In each case, the last digit of D is also the first digit of E = W+T and so we have E's first and last digits. That leaves at most 10 possible values for E, and some or all of them may not be compatible with W and the required range of T, as indicated in the following table:

W    R  T in range      E of form  Possibilities for T
------------------------------------------------------------
142 4 50 <= T <= 499 7x7 ---
152 4 50 <= T <= 499 7x7 ---
182 4 50 <= T <= 499 7x7 ---
102 5 50 <= T <= 64 9x7 ---
132 5 50 <= T <= 64 9x7 ---
128 5 65 <= T <= 499 7x5 ---
188 5 65 <= T <= 499 7x5 ---
312 5 65 <= T <= 499 9x7 ---
122 6 50 <= T <= 128 3x7 ---
192 6 50 <= T <= 128 3x7 115, 125
202 6 50 <= T <= 128 3x7 105, 115, 125
232 6 50 <= T <= 128 3x7 75, 85, 95, 105, 115, 125
268 6 50 <= T <= 128 1x5 ---
298 6 50 <= T <= 128 1x5 ---
308 6 50 <= T <= 128 1x5 ---

So we now know that N=2, S=7, R=6, and W, T, D, G, A_down, B and E are one of the combinations given by the following table:

W    T    D     G     A_down  B    E
--------------------------------------
192 115 8013 4657 36864 230 307
192 125 8003 4657 36864 250 317
202 105 8023 4637 40804 210 307
202 115 8013 4637 40804 230 317
202 125 8003 4637 40804 250 327
232 75 8053 4577 53824 150 307
232 85 8043 4577 53824 170 317
232 95 8033 4577 53824 190 327
232 105 8023 4577 53824 210 337
232 115 8013 4577 53824 230 347
232 125 8003 4577 53824 250 357

The last digit of B is 0 in every case, and it must be the same as the second last digit of D. So only the three rows with D=8003 are actually possible, and as each of those rows has the same value for B, we also know that B=250. That implies that the third digit of A_down is 2.

We know that F = S*P-R = 7*P-6 is a 2-digit number, which implies that 3 <= P <= 15. We also know that A = P!/Q is an 8-digit number that starts with 3, 4 or 5, so P! is at least such a number. That implies that P >= 11, and the fact that C = N^P+Q^N is a 4-digit number implies that N^P = 2^P <= 9999. So P is 11, 12 or 13.

If P=11, then A_across = 11!/Q = 39916800/Q is an 8-digit number starting with 3, 4 or 5. That implies that Q=1 and A_across=39916800. However, the third digit of A_across is not 2, so P=11 doesn't work.

If P=13, then A_across = 13!/Q = 6227020800/Q is an 8-digit number starting with 3, 4 or 5, which implies that Q >= 6227020800/59999999 > 100. That implies that C = N^P+Q^N > 100^2, so C is not a 4-digit number. So P=13 also doesn't work, and we can conclude that P=12.

So F = S*P-R = 78, and A_across = 12!/Q = 479001600/Q is an 8-digit number starting with 3, 4 or 5. That implies that Q is in the range RoundUp(479001600/59999999) = 8 to RoundDown(479001600/30000000) = 15. Values of Q in that range that make A_across a whole number are:

Q = 8, leading to A_across = 59875200
Q = 9, leading to A_across = 53222400
Q = 10, leading to A_across = 47900160
Q = 11, leading to A_across = 43545600
Q = 12, leading to A_across = 39916800
Q = 14, leading to A_across = 34214400
Q = 15, leading to A_across = 31933440

The third digit of A_across is 2 in the ones with Q=9 and Q=14, and C = N^P+Q^N is 4177 or 4292 respectively. That needs to intersect correctly with A_across (which it does in both cases) and also with F, which it only does in the Q=9, A_across=53222400 and C=4177 case. That implies that the first digit of A_down must be 5, which picks out the W=232, T=125 possibility of the three rows with D=8003.

That completes the solution, which has N=2, P=12, Q=9, R=6, S=7, T=125, and W=232, from which it follows that the 'across' answers are:

A = P!/Q = 12!/9 = 53222400
D = N^R*(N^S-1)-T = 2^6 * (2^7 -1) - 125 = 8003
F = S*P-R = 7*12 - 6 = 78
G = S!-N*W+1 = 7! - 2*232 + 1 = 4577

the 'down' answers are:

A = W^N = 232^2 = 53824
B = N*T = 2*125 = 250
C = N^P+Q^N = 2^12 + 9^2 = 4177
E = W+T = 232+125 = 307

And the completed grid is:

-------------------------------------------------
|A | |B | | |C | | |
| 5 | 3 | 2 | 2 | 2 | 4 | 0 | 0 |
-------------------------------------------------
| |XXXXX| |XXXXX|XXXXX| |XXXXX|XXXXX|
| 3 |XXXXX| 5 |XXXXX|XXXXX| 1 |XXXXX|XXXXX|
-------------------------------------------------
|D | | |E |XXXXX|F | |XXXXX|
| 8 | 0 | 0 | 3 |XXXXX| 7 | 8 |XXXXX|
-------------------------------------------------
| |XXXXX|XXXXX| |XXXXX| |XXXXX|XXXXX|
| 2 |XXXXX|XXXXX| 0 |XXXXX| 7 |XXXXX|XXXXX|
-------------------------------------------------
|G | | | |XXXXX|XXXXX|XXXXX|XXXXX|
| 4 | 5 | 7 | 7 |XXXXX|XXXXX|XXXXX|XXXXX|
-------------------------------------------------

Gengulphus

cinelli
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Re: Crossnumber

#167652

Postby cinelli » September 20th, 2018, 10:54 am

This is an absolute tour de force from Gengulphus. I thought this puzzle was particularly challenging and it took me ages to solve it, and I didn’t have to type out all the steps. I have been dreaming of factorials and powers of 2!

If you are looking for a hard puzzle and haven’t yet read Gengulphus’s solution, please give it a try. I promise the next one will be easier, probably a chess problem.

Cinelli

ReformedCharacter
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Re: Crossnumber

#167658

Postby ReformedCharacter » September 20th, 2018, 11:11 am

cinelli wrote:This is an absolute tour de force from Gengulphus. I thought this puzzle was particularly challenging and it took me ages to solve it, and I didn’t have to type out all the steps. I have been dreaming of factorials and powers of 2!

If you are looking for a hard puzzle and haven’t yet read Gengulphus’s solution, please give it a try. I promise the next one will be easier, probably a chess problem.

Cinelli

Thanks for the puzzle, too hard for me :)

A chess problem would be good!

RC


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