First, let's talk about electrodes.
What do the electrodes measure?
When we talk about ECG signals, we generally talk about leads - that is, difference of electrical potential between two electrodes. This voltage change comes from change in polarisation in the heart as the rise in electrical potential is created in SA node and travels to AV node before coming down and finally spreading to the ends of Purkinje fibers.
Conduction system of the heart (source)
The plotted signal you get from each lead will give you your PQRST curve, but note that even though the events are there, the plots look rather different.
Raw signals from a multi-lead setup (source)
You can see that if all you are interested in are the times and durations of the PQRST events, all you need is one lead. But which one is the best? In order to find that out, we need to talk about heart vector.
Heart vector is the sum of all the small depolarisations happening in the heart, and points roughly in the direction of where the most depolarisation is happening. During the heart cycle this vector changes in time.
Heart vector changing during the heart cycle (source)
You can see that the highest heart vector variation is on the subfigure B, which roughly lies on the line crossing an electrode on the right shoulder and the left leg.
But wait! Before you start working on your single-lead ECG patch, consider, that this will only give you the magnitude of a projection of the heart vector on the diagonal axis described in the previous paragraph. If you add one more electrode, you can build an Einthoven's triangle and actually find the heart vector (or rather it's close enough approximation).
In Einthoven's triangle method, 3 electrodes are used - 2 are placed on the shoulders and 1 on pubis, or you can place them more distally - on arms and left leg.
Einthoven's triangle (source)
This gives you 3 leads, from which you can use any two to find the heart vector (guide here).
Number of electrodes
OK, but why is it so important, that we have found the heart vector? We had to add a whole electrode to our setup!
Averaging all heart vectors occurring during a heart cycle will give us (an approximation of) mean electrical axis, which can aid in diagnosing pathological hearts.
Setups with higher electrode and lead count can give you even more insight into the heart function, but I will omit them in this answer for the sake of brevity.
3-lead ECG patch
Probably the closest to what you want to achieve is a 3-lead ECG patch described in the Quality of ECG Monitoring with a Miniature ECG Recorder (Janata, 2008). They compare the performance of a small ECG patch placed in 4 different locations to a Holter monitor for rhythm analysis. They find that the patch is useful, but there is some information lost compared to the Holter monitor. Not that the channels in the patch look really similiar compared to the ones from the Holter monitor, so placing the electrodes close together does limit the amount of information you will be able to gather.
ECG patch used in the study
Single-lead ECG arrhythmia detection
If you really are limited to to electrodes, there has been some luck in detection of arrhythmia from a single lead signal using machine learning. An example would be Detection of Atrial Fibrillation from RR Intervals and PQRST Morphology using a Neural Network Ensemble (Khamis, 2018). However, keep in mind they used a proper lead I (LA-RA) setup, not a single ECG patch.