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SUGGESTED LEVEL: UPPER SECONDARY

Below is a hierarchy of increasingly dramatic (and mathematically complicated) systems for a reality TV competition such as “Britain’s Got Talent” or “Strictly Come Dancing” to announce their competition winner. Just one rule is needed for creating each system of announcements from the previous one: in mathematics this concept is referred to as **RECURSION**.

Confused? Let’s see how this would work in practice:

**LEVEL 1 SYSTEM:** at the most basic level, we simply announce the winner. No fuss, no frills: one announcement. This contestant is usually referred to as “the winner!” but I’ll refer to them as the “level one winner” – the best kind of winner as they are the competition’s overall champion. So far so good, but let’s make things more interesting…

**LEVEL 2 SYSTEM:** rather than announcing the (level 1) competition winner, we make a series of announcements of all the contestants who are NOT the competition winner! These folks are usually referred to as “eliminated from the competition” but I’ll refer to them as “level 2 losers” – the worst kind of loser as they are out of the competition for good – commiserations. Now let’s increase the tension…

**LEVEL 3 SYSTEM:** In place of each “such and such a person is eliminated” announcement from level 2, we announce instead all the contestants who are* not eliminated!* These contestants are typically referred to as “*safely through to next week*”, but I’ll refer to them as “level 3 winners”. They’ve not won the whole competition, but they have scored a minor victory by getting through to the next round of announcements – well done! Now let’s take things one step further…

**LEVEL 4 SYSTEM: **In place of each “such and such a person is safely through to next week” announcement, we announce instead all the contestants who *are not safely through to next week* – let’s call them

We can now imagine contestants waiting anxiously to see if they are going to be the next person declared “not yet safely through to next week”. If they can avoid all the “level 4 loser” announcements until only they remain, they’ll be a level 3 winner and can now celebrate being “safely through to next week”! Now to level 5…

**LEVEL 5 SYSTEM:** In place of each “such and such a person is a level 4 loser” announcement, we announce instead all the contestants who *are not level 4 losers* – let’s call these folks

We can of course keep this idea going by announcing “level 6 losers” or “level 7 winners” and so on. But let’s pause here and examine how much work all of this will entail for our Tess Daly, Declan Donnelly or whichever presenter is making the announcements:

**LEVEL 1:** there is just one announcement to make: the competition winner. As it happens, you can find a whole bunch of ones down the first diagonal of Pascal’s Triangle (see below).

**LEVEL 2:** assuming a modest competition of, say, 8 contestants, a level 2 system will require 7 “elimination” announcements. Seven – the sum of seven ones – is the seventh **INTEGER**, (whole number), and can be found in the second diagonal of Pascal’s Triangle.

**LEVEL 3:** assuming again that we start with 8 contestants, there will be 7 “level 3 winner” announcements before just one contestant remains and is eliminated from the competition; we then announce 6 “level 3 winners” to reach the next eliminee, and so on, making a total of 7+6+5+4+3+2+1=28 level 3 announcements. 28 – the sum of the first 7 integers – is the seventh **TRIANGULAR NUMBER**, and can be found in the third diagonal of Pascal’s Triangle.

**LEVEL 4:** starting again with 8 contestants, we make 28 “level 4 loser” announcements to reach the first level 3 winner, then 21 level 4 announcements (the 6^{th} triangular number) to get our next level 3 winner, then 15, 10, 6, 3, 1, totalling 83 annoucements all in. 83 – the sum of the first 7 **triangular **numbers – is the seventh **TETRAHEDRAL NUMBER** and can be found in the fourth diagonal of Pascal’s Triangle.

**LEVEL 5: **similarly, this system would require 210 announcements, this being the seventh **PENTATOPE NUMBER**. Pentatope numbers are calculated as the sum of the first n tetrahedral numbers, and they reside in the fifth diagonal of Pascal’s Triangle.

**LEVEL 6 and above:** in subsequent systems, the total number of announcements would be the seventh “**5-simplex**”, “**6-simplex**” etc number, these residing in subsequent diagonals of Pascal’s Triangle.

If this all seems a needlessly complicated, you might consider that Level 3 systems seem to be the standard on Reality TV shows, even though a Level 1 system would be perfectly adequate for finding the overall winner. How, I wonder, would the viewing public react to a level 4 system?

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