Comparing two numbers may appear to be a simple operation. However, when those numbers are presented as arithmetic expressions, the task involves several stages of cognitive processing.
In Math Pass, the player sees two expressions, such as “21 + 15” and “7 − 39,” and must decide whether the first result is greater than, smaller than, or equal to the second. To answer correctly, the player must calculate both expressions, retain their results, compare their numerical magnitudes, and select the appropriate relation.
This structure connects Math Pass to established research in numerical cognition, particularly studies of mental arithmetic and the numerical distance effect.
In 1967, psychologists Robert Moyer and Thomas Landauer published a foundational experiment on numerical comparison. Participants were shown two digits and asked to identify the larger one as quickly as possible.
The researchers found that response time depended on the numerical distance between the values. Numbers that were far apart, such as 2 and 9, were generally compared more quickly than numbers that were close together, such as 7 and 8.
This result became known as the numerical distance effect.
One influential explanation is that numbers are represented internally as ordered magnitudes. When two values are close, their mental representations may be more difficult to distinguish. When the distance is larger, the difference is easier to identify.
Later research by Stanislas Dehaene, Emmanuel Dupoux, and Jacques Mehler examined comparisons involving two-digit numbers. Their findings showed that numerical judgments were influenced by both symbolic structure and overall numerical magnitude. This demonstrated that comparing multi-digit numbers is not always performed as a single, perfectly discrete operation.
The interpretation of the numerical distance effect continues to be debated. Nevertheless, the behavioral pattern itself—slower or less accurate comparison of nearby values—has been reproduced across many experimental tasks.
Math Pass introduces an additional requirement that is not present in a standard number-comparison experiment. The values are not shown directly. They must first be calculated.
For example, when comparing “12 + 9” with “25 − 3,” the player must:
Research on mental arithmetic has shown that the time needed to solve a problem can vary with its numerical structure and difficulty. In a classic study of simple addition, Groen and Parkman examined the relationship between the numbers in a problem and the time required to produce an answer.
Later work by Ashcraft and Battaglia provided evidence that mental addition involves both the retrieval of arithmetic information and decision processes used to evaluate a result. Larger or less familiar problems often require more processing than small, highly familiar combinations.
The exact strategy can differ between problems and individuals. A familiar expression may be answered through direct retrieval from long-term memory, while a less familiar one may require decomposition, counting, transformation, or another calculation strategy.
Because Math Pass presents two expressions, the player must keep one result available while calculating or evaluating the other. This creates a working-memory demand.
The player is not simply solving two independent equations. The results must remain accessible long enough to establish a relationship between them. If one result is forgotten or confused with an intermediate value, the final comparison may be incorrect even when the arithmetic operations were initially performed correctly.
The response stage also matters. “Greater than,” “less than,” and “equal” represent three different relationships. The player must map the comparison onto the correct response while avoiding a reversal between the left and right expressions.
Equality trials may create a distinct challenge. Instead of detecting which value is larger, the player must determine that no magnitude difference exists. This requires both calculations to be sufficiently precise and may discourage approximate judgment.
The difficulty of a Math Pass trial can be influenced by several factors:
Consider the difference between comparing results of 4 and 18 versus results of 17 and 18. Even if the arithmetic operations are equally familiar, the second comparison may demand finer numerical discrimination because the results are closer together.
Trials involving negative values introduce another layer. The player must correctly interpret numerical order below zero, where a value with a larger absolute magnitude can still be numerically smaller. For example, −32 is less than −5, even though 32 is the larger absolute quantity.
Math Pass is not identical to the experimental tasks used by Moyer, Landauer, Dehaene, or other researchers. These studies therefore do not directly prove that playing Math Pass improves general mathematical ability, working memory, or processing speed.
However, the game shares clear structural features with established numerical-cognition tasks. It requires mental calculation, temporary retention of results, symbolic magnitude comparison, relational judgment, and response selection.
As the expressions become more complex or their results become closer, the player must coordinate these processes with greater precision. The challenge does not come only from performing arithmetic. It comes from transforming two symbolic expressions into numerical values and then deciding how those values relate to one another.
Determining whether repeated Math Path practice produces improvements that transfer beyond the game would require a controlled study with appropriate comparison groups and independent mathematical assessments. What can be stated more confidently is that every trial engages a sequence of processes that has been studied extensively in experimental research on mental arithmetic and numerical comparison.
Moyer, R. S., & Landauer, T. K. (1967). Time required for judgements of numerical inequality. Nature, 215, 1519–1520. https://doi.org/10.1038/2151519a0
Dehaene, S., Dupoux, E., & Mehler, J. (1990). Is numerical comparison digital? Analogical and symbolic effects in two-digit number comparison. Journal of Experimental Psychology: Human Perception and Performance, 16(3), 626–641. https://doi.org/10.1037/0096-1523.16.3.626
Groen, G. J., & Parkman, J. M. (1972). A chronometric analysis of simple addition. Psychological Review, 79(4), 329–343. https://doi.org/10.1037/h0032950
Ashcraft, M. H., & Battaglia, J. (1978). Cognitive arithmetic: Evidence for retrieval and decision processes in mental addition. Journal of Experimental Psychology: Human Learning and Memory, 4(5), 527–538. https://doi.org/10.1037/0278-7393.4.5.527