While surfing the webpages of a variety of newspapers this morning, I stumbled on the following….

On the staged-reality-tv show Big Brother (UK version), they gave the housemates a mathematical task: they had to compute three different sums, then use the three resulting answers as the combination to a safe.

Implicit in the problem was that the three calculations should each result in a two-digit integer.

Video posted to the Channel 4 website shows the contestants muttering, struggling, and having an extremely difficult time of it. And with good reason!

Here are the three calculations they were given, as posted on the Channel 4 website:

Sum#1: 3 x 17 – 24 + 78 x 9 ÷ 5 – (13²) + (65 – 29) ÷ 4 + (4²) – (7 x 3) + (3²) + 99 – (7²) – 49

Sum#2: 1396 x 2 ÷ 4 — (12²) + 46 x 2 ÷ 40 x (5²) – (7 x 99) x 3 – (11²) x 5 – 219

Sum#3: 100 – 33 x 5 + 665 ÷ (5²) x 17 – 248 x 3 ÷ (4²) + 52 ÷ 7 + (273 – 217)

From the look of things on the on-line video, I’m guessing that the contestants had no writing implements, and had to do all of this in their head. That makes this challenging enough, I suppose.

Making matters worse is that none of the three sums is integer valued; they work out to 62/5, -4583/2, and ~~26189/70~~ 28289/70. Rather, these are their values if one computes using the usual order of operations, where exponents have precedence over other operations, where multiplication and division take precedence over addition and subtraction, where calculations are performed left-to-right, and parentheses can be used to override this sequencing. (“PEMDAS” is a popular acronym with my students, standing for “Parentheses, Exponents, Multiplication and Division, Addition and Subtraction”, and sometimes recalled using the mnemonic “Please Excuse My Dear Aunt Sally”)

Apparently the folk who created this puzzle expected their contestants to work left-to-right, ignoring operator precedence, in the way that a $1 calculator might do. (Calculators that do pay heed to order of operation conventions are often marketed as “scientific” calculators.)

For example, the first sum *should *go as follows:

3 x 17 – 24 + 78 x 9 ÷ 5 – (13²) + (65 – 29) ÷ 4 + (4²) – (7 x 3) + (3²) + 99 – (7²) – 49

= 3 x 17 – 24 + 78 x 9 ÷ 5 – 169 + 36÷ 4 + 16 – 21 + 9 + 99 – 49 – 49

= 51 – 24 + 702/5 – 169 + 9 + 16 – 21 + 9 + 99 – 49 – 49

= 62/5

But I suspect the intended calculation was instead:

3×17 = 51, 51-24 = 27, 27+78 = 105, 105×9 = 945

945÷5=189, 189-(13²)=20, 20+(65-29)=56. 56÷4=14,

14+(4²)=30, 30-(7 x 3)=9, 9+(3²) + 99 – (7²) – 49 = 19

Similar (incorrect!) computations for sum #2 and sum #3 yield 31 and 75, respectively.

Clearly it is important that we agree on our order of operations. But why do we prefer one over the other? Is this merely a cultural convention? One stock answer to this is to point to the algebra of polynomials: our conventions regarding operator precedence play a central role in how we interpret linear equations (e.g. what is the slope of the line ?), how we interpret polynomials (e.g. is a quadratic or a cubic polynomial?), and how we compute sums and products of polynomials.

But this morning, having not yet had my first cup of coffee, I wonder: is it possible to change the rules of arithmetic, so that all operations have the same precedence (unless exceptions are forced by parentheses), and to develop a meaningful algebra based on similar principles? It seems to me that the answer is yes, and I wonder exactly what is lost by doing so, other than familiarity.