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Sandwiches

Stage: 2, 3, 4 and 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Thank you Tim from Gravesend Grammar School for your solution.

The solutions are 312132 and its mirror image 231213. We call these 3-sandwiches.

You cannot do the same with only ones and twos, because between the two twos there must be two digits, which have to be ones as these are the only available digits, but that means that between the two ones there are no digits, so this is not possible.

The only 4-sandwiches are 41312432 (and its mirror image).

You will probably have found quite a lot of 7-sandwiches because there are altogether 26 (and their mirror images).

You may have tried, and failed, to find 5-sandwiches and 6-sandwiches but there aren't any. If you have tried to find 8-sandwiches you will probably have had more success as there are altogether 150 of these (and their mirror images).

See 'Impossible Sandwiches' for a simple proof that n-sandwiches can only be made when n=4m or n=4m-1 (for integers n and m). For example solutions do not exist for n=5 or n=6 but solutions do exist for n=7 and 8.

The following BASIC program gives the 4-sandwiches. You could try making some small changes to get the 7-sandwiches. Alternatively you could take up the challenge of finding all the solutions without using a computer, or write a computer program in a different language to find the solutions.

10      DIM A(8)
 
20      FOR A(1)= 1 TO 6:       REM A(1) and A(5) are positions for 1's
 
30      FOR A(2)= 1 TO 5:       REM A(2) and A(6) are positions for 2's
 
40      FOR A(3)= 1 TO 4:       REM A(3) and A(7) are positions for 3's
 
50      FOR A(4)= 1 TO 3        REM A(4) and A(8) are positions for 4's
 
60      FOR T=1 TO 4
 
70      A(T+4)= A(T) + T + 1:   REM Second positions of 1 to 4 spaced correctly from the first
 
80      NEXT
 
90      PROCCHECK
 
100     IF Z = 1 PROCPRINT
 
110     NEXT: NEXT: NEXT: NEXT
 
120     END
 

 
130     DEF PROCCHECK
 
140     Z=1:            REM Z=0 eliminates cases where two numbers occur in same position
 
150     FOR I=1 TO 7    
 
160     FOR J=I+1 TO 8  
 
170     IF Z=0 GOTO 200 
 
180     IF A(I)=A(J) THEN Z=0
 
190     NEXT: NEXT
 
200     ENDPROC
 

 
210     DEF PROCPRINT
 
220     FOR M=1 TO 8
 
230     FOR K=1 TO 8
 
240     IF A(K)=M AND K < 5 PRINT K;
 
250     IF A(K)=M AND K > 4 PRINT K- 4;
 
260     NEXT: NEXT
 
270     PRINT
 
280     ENDPROC
 

There are no N-sandwiches for N=1, 2, 5, 6 or 9 or any number that leaves a remainder of 1 or 2 when divided by 4. (Click here for a proof of this fact.)