This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
This activity investigates how you might make squares and pentominoes from Polydron.
If you had 36 cubes, what different cuboids could you make?
Thank you Tim from Gravesend Grammar School for your
The solutions are 312132 and its mirror image 231213. We call
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).
'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
100 IF Z = 1 PROCPRINT
110 NEXT: NEXT: NEXT: NEXT
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
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
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.)