On 15 jul, 12:15, "Harald Schilly" <harald.schi...@gmail.com> wrote:

Hi. There is a simple saying: ““If you can count to ten you can count
to infinity really” to motivate six year olds. Let’s now move on by
motivating all ages using SAGE. I just give examples and not lines of
code, mainly because to promote the usage of SAGE and to encourage
others to check the results found. {Note: I encountered several bugs
in SAGE, but release 3.1 should be fine}
1.      After counting we learn our youngsters to use "*","+","-". Not
always easy to find motivating exercises. The following question
proved to be fun for either an individual or a group. Especially in
cases where people had to wait for one hour or longer, e.g. bus trip
et cetera.
a.      Easy start. Take the for instance the numbers 3,5,7,8. The
challenge is:  make as many numbers as possible (i) starting from 0,
(ii) you may use each number only once, and (iii) using the three
operands "*","+","-".
The amazing fact is that you can produce consecutive numbers in the
range 0..65. The following output comes from a SAGE notebook:
1 = 3+5-7 , 2 = 3+7-8 , 3 = 3 , 4 = 3+8-7 , 5 = 5 , 6 = 3+8-5 , 7 =
7 , 8 = 8 , 9 = 3+5-7+8 , 10 = 3+7 , 11 = 3+8 , 12 = 3*8-(5+7) , 13 =
3+7-5+8 , 14 = 3*5+7-8 , 15 = 3*5 , 16 = 3*5-7+8 , 17 = 3*8-7 , 18 =
3*7+5-8 , 19 = 3*8-5 , 20 = (3+8-7)*5 , 21 = 3*7 , 22 = 3*5+7 , 23 =
3+5+7+8 , 24 = 3*7-5+8 , 25 = (8-3)*5 , 26 = 3*7+5 , 27 = 5*7-8 , 28 =
(5+7)*3-8 , 29 = 3*7+8 , 30 = 3*5+7+8 , 31 = 3*8+7 , 32 = 5*7-3 , 33 =
5*8-7 , 34 = 3*7+5+8 , 35 = 5*7 , 36 = 3*8+5+7 , 37 = 5*8-3 , 38 =
5*7+3 , 39 = (5+8)*3 , 40 = (3+7-5)*8 , 41 = 7*8-3*5 , 42 =
(3+7)*5-8 , 43 = 5*7+8 , 44 = (5+7)*3+8 , 45 = (7+8)*3 , 46 =
3+8+5*7 , 47 = 5*8+7 , 48 = (3+5)*7-8 , 49 = (3*5-8)*7 , 50 =
(3+7)*5 , 51 = 7*8-5 , 52 = (5+8)*(7-3) , 53 = 7*8-3 , 54 = 7*8+3-5 ,
55 = (3+8)*5 , 56 = (3+5)*7 , 57 = (3+5)*8-7 , 58 = (3+7)*5+8 , 59 =
3*8+5*7 , 60 = (5+7)*(8-3) , 61 = 3*7+5*8 , 62 = (3+8)*5+7 , 63 =
3*7*(8-5) , 64 = (3*5-7)*8 , 65 = (3*7-8)*5 , 66 has no expression
b.      Less easy.  Is 3,5,7,8 the best 4-tuple? Two ways to answer this
using SAGE:
i.      All (reasonable) combinations of 4 different numbers
ii.     Hard: Prove it by (a) produce symbolic expressions – there are
many! – and (b) just try to solve them.
c.      You can play around by also allowing other operands like “/”, “^”,
“mod or % in SAGE”, or you can allow also negative numbers and look
for the longest consecutive row. In the case of 3,5,7,8 and the three
operands "*","+","-" I found 94 consecutive numbers.
d.      If you add “^” (so 2^3=8), you find that the 4-tuple 2,3,7,8
delivers a nice row of numbers 0..75 as the following SAGE output
shows:
1 = 2**3-7 , 2 = 2 , 3 = 3 , 4 = 2+3+7-8 , 5 = 2*3+7-8 , 6 = 2*3 , 7 =
7 , 8 = 8 , 9 = 2**3-7+8 , 10 = 2+8 , 11 = 2*7-3 , 12 = 2+3+7 , 13 =
2*3+7 , 14 = 2*3+8 , 15 = 2**3+7 , 16 = 2**3+8 , 17 = 2*7+3 , 18 =
(2*7-8)*3 , 19 = 2*7-3+8 , 20 = 2+3+7+8 , 21 = 2*3+7+8 , 22 = 2*7+8 ,
23 = 2**3+7+8 , 24 = 3**2+7+8 , 25 = 2*7+3+8 , 26 = 2*8+3+7 , 27 =
(2+3)*7-8 , 28 = (3+7)*2+8 , 29 = 3*7+8 , 30 = (2+8)*3 , 31 =
2+8+3*7 , 32 = (7-3)*8 , 33 = (2+3)*8-7 , 34 = 2*3*7-8 , 35 =
(2+3)*7 , 36 = (3+7+8)*2 , 37 = 2*8+3*7 , 38 = 2*7+3*8 , 39 =
(7-2+8)*3 , 40 = (2+3)*8 , 41 = 2*3*8-7 , 42 = 2*3*7 , 43 =
(2+3)*7+8 , 44 = 7**2+3-8 , 45 = (2+7)*(8-3) , 46 = 7**2-3 , 47 =
(2+3)*8+7 , 48 = (2**3)*7-8 , 49 = (2+8-3)*7 , 50 = 2*3*7+8 , 51 =
(2+7+8)*3 , 52 = 7**2+3 , 53 = 7*8-3 , 54 = 7**2-3+8 , 55 = 2*3*8+7 ,
56 = (2**3)*7 , 57 = (2**3)*8-7 , 58 = (3*7+8)*2 , 59 = 7*8+3 , 60 =
(3+7)*(8-2) , 61 = 2+3+7*8 , 62 = 2*3+7*8 , 63 = (3**2)*7 , 64 =
(2**3)*7+8 , 65 = 3**2+7*8 , 66 = (2*7+8)*3 , 67 = (2+8)*7-3 , 68 =
7-3+8**2 , 69 = (2+7)*8-3 , 70 = 2*7*(8-3) , 71 = (2**3)*8+7 , 72 =
(2+7)*8 , 73 = (2+8)*7+3 , 74 = 3+7+8**2 , 75 = (2+3)*(7+8) , 76 has
no expression
e.      Allowing negative numbers, I found that 103 consecutive numbers can
be produced. Such an exercise proved to be fun with motivated adults
and some good drinks.

2.      Playing around with four number is relatively easy. More difficult
is the 5-tuple question, even in the simple case by only allowing the
operands  "*","+","-". The reason is mainly because the number of
possibilities ‘explodes’.
To conclude: SAGE helped me to find relative simple exercises and
questions to motivate people of all ages (i) to do exercises “using
pen and paper”, and (ii) to become aware of the depth of even simple
mathematics. I learned from SAGE that “mathematics on a computer is
even better then gaming”. Roland