Golf Competition Conundrum

Slow play is once again in the headlines. On this occasion the final round of the rain-interrupted 2019 USPGA Genesis Open, where golf commentators are openly discussing the pace of play of the winner J.B. Holmes rather than the quality of his golf.

Golfing Herald’s take on the pace of play is expressed through ‘Competition Conundrum’, a fictional golfing story loosely based on the classic Aesop fable of ‘The Tortoise and the Hare’, whose moral is that you can be more successful by doing things slowly and steadily as opposed to acting quickly and carelessly.

In this tale, Golfing Herald illustrates that a 4-ball format can be quicker than a 3-ball format which in turn can be quicker than a 2-ball format!!

Conundrum

The annual renewal of the ‘Play Ready Golf Salver’ open competition, hosted by the Fictional Golf and Country Club on the first Saturday of July was fast approaching. However, for this year’s renewal of the prestigious individual medal competition, the Competition Committee had been set the following unexpected conundrum by the Club President, namely:

“The draw for the ‘Play Ready Golf Salver’ must ensure that the final group complete their 18 holes by the earliest possible time”.

The chairman of the committee enquired of the Club President whether there were variables and/or assumptions which needed to be considered by himself and his fellow committee members to assist in addressing and subsequently resolving this most unusual and curious conundrum.

Begrudgingly, the Club President articulated that:

1. All players will start at the 1st.
2. The first group will tee off at 08:00.
3. Allow a 10-minute interval between each group.
4. Assume a 2-ball will complete their round in 3 hours.
5. Assume a 3-ball will complete their round in 3 and 1/2 hours.
6. Assume a 4-ball will complete their round in 4 hours.
7. There will be a minimum of 17 players and a maximum of 96 players.

At this juncture, please don’t think (or expect!!) that what follows is narrative that is layered with complex Mathematical theorems (though there are some very simple but unavoidable mathematical summations).

Fingers crossed (ala Harry Vardon…not original if you have read the article ‘Honesty and Honestly’) you will enjoy the rest of this fictitious story.

For the record, I even surprised myself with what followed as I attempted to resolve the conundrum set by the Club President!!

Emergency Committee Meeting

The following day, the Chairman called an Emergency Committee Meeting (ECM) with this most peculiar conundrum as the only item on the agenda.

He and his fellow committee members then convened for the ECM later that day in the clubhouse, where he articulated the conundrum and the underpinning parameters.

First Hare

Almost immediately, the Deputy Chairman claimed in his customary excitable manner that he had the answer.

“Every golfer knows that a 2-ball format is the fastest. If the Salver receives an even number of entries then all groups will play in a 2-ball format. If an odd number of entries are received then all groups will play in 2-ball format apart from the final group who will play in 3-ball format”.

Following his succinct response to the conundrum, several members of the committee nodded silently their individual and collective approval.

Second Hare

The Treasurer then proceeded to get up on his hind legs and inform the Deputy Chairman that as usual, he had absolutely no idea what he was talking about. “We always play the Salver as a 3-ball format augmented by either a 1 x 2-ball group or 2 x 2-ball groups to cater for all permutations of entries”. After this rather assertive retort, he returned to his seat in a slow and deliberate manner whilst an uncomfortable tension quickly descended over proceedings.

Tortoise

The Chairman attempted to restore some semblance of order by proposing a show of hands as that was probably, in his opinion, the most expedient method to decide between the respective solutions articulated by the Deputy Chairman and the Treasurer.

Unfortunately for the Chairman, this did not resolve the impasse as he had hoped as the show of hands resulted in a tie and an abstention from Mr G Herald, who had only recently joined the committee, filling the vacancy caused by the sudden death of one of the founding fathers of the Fictional Golf and Country Club.

All eyes were now focused on Herald to explain why he had abstained. After a few nervous gulps of water, he explained the reason for his abstention. “Mr Chairman and fellow committee members. The Deputy Chairman is not correct whilst the Treasurer is only partially correct and that is why I abstained. However, I would welcome the opportunity to present what I believe is the answer to the conundrum”.

After the not to be unexpected mutterings such as “How can I not be correct”, “Who the hell does he think…” and “Whippersnapper”, all peppered with a cocktail of industrial-strength expletives, the Chairman invited Mr Herald to explain all.

What now follows is an abridged summary of his logical argument and explanation (with interruptions, questions, puzzled looks, etc. airbrushed for the purposes of expediency).

48 Entries

Mr Herald commenced by saying that this conundrum is resolvable, unlike the famous challenge set by “The Seven Bridges of Königsberg” (have a look at Wikipedia to read about a puzzle that started as an alcohol-fuelled drinking game in darkest Prussia in the early 18th century!!).

The crux of the solution is to optimise the number of groupings allied with the (assumed) elapsed time of the final grouping.

Let’s “tee off” by assuming the Club receives exactly 48 entries for this year’s Salver. This could result in 24 x 2-ball, 16 x 3-ball or 12 x 4-ball formats (all the other format combinations have been discounted as they can be proven to not resolve the conundrum). The following summarises each possible option:

Format	First Group (Tee Time)	Final Group (Tee Time)	Final Group (Finish Time)
2-ball	08:00	11:50	14:50
3-ball	08:00	10:30	14:00
4-ball	08:00	09:50	13:50

Thus the conclusion reached, based on exactly 48 entries and meeting the Club Presidents directive, is that the Salver would have to be played in a 4-ball format. Thus, at least for this specific number of entries, a 4-ball format is quicker than a 3-ball format which in turn is quicker than a 2-ball format!!

49 to 96 Entries

Let’s now look at the Club receiving between 49 and 96 entries inclusive for the Salver. Does the argument still hold true that a 4-ball format is the quickest format? Yes, it does. In fact, the gap between the finish times of the respective formats continues to widen so that if the Salver is fully subscribed we have the following:

Format	First Group (Tee Time)	Final Group (Tee Time)	Final Group (Finish Time)
2-ball	08:00	15:50	18:50
3-ball	08:00	13:10	16:40
4-ball	08:00	11:50	15:50

What happens when the number of entries in this range is not exactly divisible by 4? Not an issue as:

• If remainder = 1, then 1 x 2-ball and 1 x 3-ball are drawn before all the 4-ball groups are drawn.
• If remainder = 2, then 1 x 2-ball is drawn for the first group before all the 4-ball groups are drawn.
• If remainder = 3, then 1 x 3-ball is drawn for the first group before all the 4-ball groups are drawn.

34 to 47 Entries

This subset of possible entries for the Salver generates some very interesting and at the outset of embarking on this article some unexpected results.

For this specific range, the argument for the 4-ball format being the quickest format as expressed above for the range 48 to 96 remains constant and true in that no other combination can achieve an earlier finish time. However, several entries in this range can achieve a matching finish time (examples include 35 and 37) when aligned to the 3-ball format solution proposed by the Treasurer.

However, if the Salver received exactly 40 entries, the 4-ball format is quicker than the 3-ball format by 10 minutes, so to simplify the overall solution to the conundrum, if the Salver receives at least 34 entries then the 4-ball format must be adopted.

Half-Way House

If you have not resorted in utter desperation to taking a mashie niblick or an equivalent instrument of destruction to your laptop or mobile, then at the notional half-way house, the 4-ball format must be adopted for the Salver to resolve the conundrum if there are between 34 and 96 entries!!

17 to 33 Entries

Unlike the previous ranges, the 4-ball format is no longer a factor (though I have excluded the underpinning mathematics) so it comes down to a straight golfing shootout between the Deputy Chairman’s 2-ball format and the Treasurers 3-ball format to complete resolution of the Club President’s conundrum.

Several entries in this range result in a tie between the respective formats but that is as good as it gets for the Deputy Chairman. Thus, to continue the tenor of simplicity the 3-ball format proposed by the Treasurer must be adopted for this specific range and that is why the Treasurer was partially correct with his solution.

Closing the Meeting

After Herald had finished, the Chairman thanked him for articulating such a cohesive and ultimately a simple solution to the conundrum. After a further show of hands, the members of the committee all supported ‘Heralds Solution’ apart from the Deputy Chairman, who petulantly offered his resignation from the committee. The Chairman thanked him for all his input and efforts over the years and accepted his resignation.

The Chairman informed the committee that without further delay he would advise the Club President of the solution and then he closed the ECM.

Postscript

Of course, there are more holes in this fictitious yarn than you will find at say the historic golf resort of Pinehurst, North Carolina. There could well be some glaring mistakes with the mathematics so that the article effectively becomes null and void.

If not, then maybe some food for thought for Competition Committees all around the world.

To finish, the selection of 17 entries as the lower limit was not an arbitrary choice, as the wonders and magic of the numbers 2 to 16 inclusive played havoc with achieving a neat and simple resolution to the conundrum. For completeness, the table below illustrates the wonders of mathematics in this lower range.

# Entries	# 2-ball Groups	# 3-ball Groups	Whose Solution?
2	1	0	Deputy Chairman
3	0	1	Treasurer
4	2	0	Deputy Chairman
5	1	1	Both!!
6	3	0	Deputy Chairman
7	2	1	Deputy Chairman
8	4	0	Deputy Chairman
9	0	3	Treasurer
10	5	0	Deputy Chairman
11	1	3	Treasurer
12	6	0	Deputy Chairman
13	2	3	Treasurer
14	7	0	Deputy Chairman
15	0	5	Treasurer
16	8	0	Deputy Chairman