One way in to the Ring on a String investigation:

Two Excel sheets are offered:
A straightforward spreadsheet construction.

Or a solution with more features.

Both files use the same basic approach, but the "more features" sheet uses conditional formatting, the MIN function, increment buttons and an inset graph, to assist presentation.

In the simpler sheet the length for AD is in cell C2 and the length for BC is in cell E2.
These values will later be changed to see the effect this has on the minimum value of the string length DRC.

The minimum length of DRC indicates the actual ring's position.

DR is calculated by using Pythagoras' Theorem with the distance AD and a distance along the string.

Different ring positions along the string will be examined, starting at 0 and going up to 100, in steps of 1.

Hence column B contains the values 0 to 100.

0 has been entered directly into B5 , but B6 contains the formula: =B5+1 , so that the formula can then be copied (replicated) downwards as far as necessary.

In this case until it gives a value of 100.

The following formulae might need a little explaining:

Cell C5 contains the formula for the length DR
=SQRT($C$2^2+B5^2)

Cell D5 contains the formula for the length RC
=SQRT($E$2^2+(100-B5)^2)

SQRT is the square root function.

Where ^2 appears in the formula it means squared.

In the C5 formula, C2 is the distance AD
(But is written $C$2 so that when this formula is copied (replicated) to new cells it will keep C2 as an absolute reference and not adjust it to whichever cell is in the same relative position as the position of C2 in the original formula.)

B5 is the distance along the string, and represents the ring position being tested in that row of the sheet.

In the D5 formula, the same thing is happening but (100-B5) replaces B5 because the length used when Pythagoras' Theorem is applied will be the remainder of the string, from that ring position to the end at 100 cm.

E5 holds the formula C5+D5 .
The minimum value of this total, as B5 goes from 0 to 100, indicates the actual ring's position for the chosen AD and BC values.

By changing the values of AD and BC can you find a relationship between these choices and the position of the ring?
When you discover a relationship try to account for why that has to be the relationship connecting AD, BC and the ring's position.

The second sheet, using extra features of Excel, does exactly the same job but with some refinements:

• Increment buttons are used to change the values of AD and BC
  
• Conditional formatting has been used to make conspicuous the calculation results which are of most interest.
  
• The actual minimum value can be reported automatically by Excel using the MIN function. So this is in view right at the top, along with the AD and BC choices.
  
• The VLOOKUP function has been used to report the distance of the ring from A that produced the minimum length DR+RC.
  and finally:
• A simple XY graph has been inserted to illustrate the ring's position for each choice of values for AD and BC.

Fuller information about each of these techniques will appear on this site in the next month or two.