Problems & Puzzles: Puzzles

Puzzle 94The Domino & the primes

For this puzzle maybe is a good idea you to get a Domino (game).

For the purpose of our questions you can put together the 28 dominoes (pieces) in two type of arrangements:

Arrangement A (just showing 3 pieces):

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to produce one number as concatenations of all the numbers

Arrangement B (just showing 3 pieces)

 ? ? ? ? ? ?

to produce two numbers, one number as a concatenations of all the numbers of the superior row and the other number as a concatenation of all the numbers of the inferior row.

Questions:

1. Get the largest prime number possible P, using the most of the 28 pieces according to arrangement A, without worrying for any matching between contiguous pieces (as in the normal domino game)

2. Redo the exercise 1 keeping the matching between contiguous pieces as in the normal domino game.

3. Redo exercises 1 & 2 getting the least prime numbers.

4. Get the largest two prime numbers P1 & P2, using the 28 pieces according to the arrangement B, such that (P1-P2) is minimal

5. Redo the exercise 4 getting the least two prime numbers P1 & P2, using the 28 pieces according to the arrangement B, such that (P1-P2) is minimal

Solution

Jeff Burch sent (June 4, 2000) the following solution to question 1:

"Using all 28 dominos won't yield a prime number because the sum of all numbers on all the dominoes is 168. All numbers like this will be divisible by 3. Using 27 dominos, omitting the one with one zero and one one the largest prime in problem #1 is 666564636261605554535251504443424140333231302220120011...In the Maple V Release 4 program the isprime function when tested with this number returns 'true'. This means the number is prime"

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Jeff Heleen, found (16/06/2000) another solution to question 1 but using another version of Domino:

"...Some domino sets have 55 tiles, going up to double nines. Using a set like this, the largest prime (for question 1) I have found so far is:

999897969594939291908887868584838281807776757473727170666564/
636261605554535251504443424140333231302211002021

omitting only the 0/1 tile...I produced it by trial and error, switching tiles at the end, till one turned up prime. I used UBASIC's ECM to test it."

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Giuliano Daddario sent (22/8/01)the following solution to question 3.1:

000102030405061112131415162223242526333435444555566663

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Later (26/8/2001) he sent his solution to the question 3.2:

the least prime number possible P, using the most of the 28 pieces according to arrangement A, keeping the matching between contiguous pieces as in the normal domino game is (eliminating only the piece 0-1)

000221111330044115522223333442266335555444466005566661

I would explain my line of reasoning about puzzle 94. Using all the 28 pieces, if the number starts with the piece 0-0, it has to terminate with a piece like A-0 (otherwise one of the zeroes is unmatched). Now we have to eliminate a piece like X-0 or 0-X (otherwise we have to put one zero in last position, and the number is not prime). After the elimination, to guarantee the matching of all the "X", the last digit has to be X. So, X=1 and you have to eliminate the piece 0-1 (we can't eliminate the piece 0-3, because in that case the number is divisible by 3).

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The Monday 27/8/2001, Giuliano wrote:

666555544663355226600551144443333224400003311110022221, this is the answer to the question 2.

the solution of part 4 is:

6543210601021032104320102433
6543210566565465436542213541

The solution to the part 5 is:

0123456100101201230243125433
0123456065645634562465236541

and their difference is 34455566667777888892.

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If Giuliano's solutions are OK, this game is over...

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