There is a puzzle called a number pyramid, where you add each adjacent pair and stack the sums upward. Solve one in your head and numbers go missing. You hold the sums from the row below, start on the next row, and the number you were certain you had is gone. Every individual addition is trivial and you still can't finish.
That isn't an arithmetic problem. It's the capacity of working memory. Here is what I found when I looked into how much it holds and whether training increases it.
What a number pyramid is
Put a row of numbers along the bottom, write the sum of each adjacent pair on the row above, and repeat until you reach the apex. They turn up on primary-school worksheets and in mental arithmetic apps.
A bottom row of 3 1 4 2 gives 4 5 6, then 9 11, then 20. As a puzzle, some cells are blanked and you recover them from the rest: addition when the two below are known, subtraction when the cell above and one below are known.
Everything that follows assumes you are solving one without writing anything down.
What working memory is
It gets used interchangeably with short-term memory, but the two are distinguished. Short-term memory refers to holding something; working memory refers to holding it while operating on it. Repeating a phone number back is the first. Adding up its last four digits is the second.
The model most people reach for is the multi-component one Baddeley and Hitch proposed in 1974, which posits three parts:
- The phonological loop: holds words and numbers as sound. Repeating a number under your breath so you don't lose it happens here
- The visuospatial sketchpad: holds shape and position. Remembering which cell of the pyramid was blank happens here
- The central executive: decides where attention goes, and performs the operation itself
Mental arithmetic uses all three. You hold numbers as sound, track where on the grid you are, and carry out the addition. Congestion in any one of them stops the whole thing.
Where 7±2 came from
The number everyone knows for working memory capacity is seven plus or minus two. It comes from a 1956 paper by Miller, titled after that figure.
What it counts is not items but chunks: units of meaning. 090-1234-5678 is eleven digits but three chunks if you learn it in those groups. The same digits as 09012345678 get abruptly harder because the cues for chunking are gone.
So what is the real figure
Seven plus or minus two became famous enough to outlive the estimate. Later work puts the number lower.
The tasks Miller drew on allowed rehearsal, the silent repetition that refreshes an item, and allowed chunking across items. Measure again with both suppressed and what survives is three to five items, centred around four (Cowan, 2001). That is the figure to reason from.
It matches how the arithmetic feels. Once a four-row pyramid has you holding four or more intermediate values, you start dropping them.
How big a chunk is
Capacity may be four chunks, but the contents of a chunk differ from person to person.
The well-known demonstration is the chess study Chase and Simon reported in 1973. Strong players shown a position from a real game for a few seconds could reconstruct almost all of it, yet on a board with pieces scattered at random they did no better than novices. Their capacity had not grown. A meaningful arrangement had simply become one unit.
Arithmetic works the same way. Someone who retrieves 7 + 8 as 15 rather than computing it spends no working memory there, in the same way that knowing your times tables changes the cost of long multiplication. This, rather than raw capacity, is usually what improves with practice.
What the arithmetic costs
Mental arithmetic is heavy because holding and operating happen at once.
Take a pyramid solved from the bottom up. You produce three sums on the second row, then start the third row while still holding all three. Nothing can be discarded until it has been consumed. Each row up holds fewer values, so the later stages actually get easier.
The first move is the hard one. Going from four bottom-row values into the second row is where the number held at once peaks.
A carry makes a step heavier again. A pair like 7 + 8 that carries, when the answer can't simply be retrieved, means running the make-ten-then-add procedure, which occupies the central executive. Whatever you were holding is most likely to slip right then.
The peak number of values held at once explains perceived difficulty rather well. I use that quantity as a difficulty axis, which is covered in generating number pyramids.
Can it be trained
This is the question that gets asked, and answering it in one piece makes it inaccurate.
The trained task itself improves. Do n-back every day and your n-back score rises. That much has been reproduced repeatedly.
The issue is what happens beyond it. In 2008 Jaeggi and colleagues reported that n-back training raised fluid intelligence, which drew a great deal of attention. Replications did not reproduce the result, and the conclusion from the meta-analyses is that transfer to tasks resembling the trained one (near transfer) does occur, while transfer to intelligence measures or everyday work (far transfer) has not been established (Melby-Lervåg & Hulme, 2013, among others).
Commercial claims drew regulatory attention as well. In 2016 the US FTC acted against Lumosity over advertising that promised improved cognition and protection against dementia without adequate support, settling for a payment of two million dollars.
Unestablished far transfer is not the same thing as demonstrated absence of an effect. Measurement here is genuinely hard and parts of the debate remain open. Still, at present there is no basis for saying that doing this will make you smarter, and it is safer to proceed on that assumption.
Useful as a measure of load
As a way to train, it can't be promised. As a yardstick for load, it works.
Plenty of mental arithmetic apps set difficulty by row count, which turns out not to track much. A four-row puzzle full of subtractions and carries beats a five-row addition-only one comfortably. Counting the peak number of values that must be held at once produces an ordering much closer to how the puzzles feel.
Pyramid Mental Arithmetic is built with that quantity as its difficulty axis. It does not claim to make anyone smarter. I built it because the puzzle is good.