For a recreational purposes I'm fine tuning my wall clock sheet and like to ask about tips how to esthetically modify the summation function for 10, 11 and 12. Below is the image of the final result:
I have checked sigmas in wolframalpha so that they are correct:
1: Sum[k^1, {k, 1, 1}]
2: Sum[k, {k, -1, 2}]
3: Sum[k, {k, 1, 2}]
4: Sum[k^1 + k, {k, -1, 2}]
5: Sum[k^k, {k, 1, 2}]
6: Sum[k + k, {k, 1, 2}]
7: Sum[(-k)^k k, {k, 1, 2}]
8: Sum[k^k + k, {k, 1, 2}]
9: Sum[k^k k, {k, 1, 2}]
10: Sum[k^k + k^k, {k, 1, 2}]
11: Sum[k/k, {k, 1, 11}]
12: Sum[k^k k + k, {k, 1, 2}]
And now I'd like to know, if it is possible to meet these requirements on the last 3 numbers:
- maximum of three occurrences of the index (k or i in the picture) can be used. 10AM and 12PM uses four...
- maximum of three 1's can be used. now 11AM is not esthetically meeting this criteria since 11 is not 1 nor 1+1 nor 1.1
- but yes, decimals can be used, but all 1's are counted as separate digits then. 1.1 will take two 1's, so one 1 is left.
I have no other strict specifications, but as said esthetics and other way intriguing solutions may pass on table so criterias may change based on solutions.
Other clock sheets made year back may give few more ideas: https://www.pinterest.com/markomanninen/math/
For 10, $\sum_{i=1}^{1+1}i(i+i)$ has the same effect with one fewer $i$.
For 11, $\sum_{i=1-1}^{1+1}(i+i)^i$ violates the "three 1s" rule instead (and requires the fairly common $0^0=1$ convention), but you may like it better than explicitly using 11.
For 12, $\sum_{i=-1}^{1+1}i(i+i)$ works.