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Edited by: Ann Dowker, University of Oxford, United Kingdom

Reviewed by: Matthias Hartmann, Universität Potsdam, Germany; Koen Luwel, KU Leuven, Belgium

This article was submitted to Developmental Psychology, a section of the journal Frontiers in Psychology

†These authors have contributed equally to this work.

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Numerical categories such as parity, i.e., being odd or even, have frequently been shown to influence how particular numbers are processed. Mathematically, number parity is defined categorically. So far, cognitive, and psychological accounts have followed the mathematical definition and defined parity as a categorical psychological representation as well. In this manuscript, we wish to test the alternative account that cognitively, parity is represented in a more gradual manner such that some numbers are represented as “more odd” or “more even” than other odd or even numbers, respectively. Specifically, parity processing might be influenced by more specific properties such as whether a number is a prime, a square number, a power of 2, part of a multiplication table, divisible by 4 or by 5, and many others. We suggest that these properties can influence the psychologically represented parity of a number, making it more or less prototypical for odd- or evenness. In the present study, we tested the influence of these numerical properties in a bimanual parity judgment task with auditorily presented two-digit numbers. Additionally, we further investigated the interaction of these numerical properties with linguistic factors in three language groups (English, German, and Polish). Results show significant effects on reaction times of the congruity of parity status between decade and unit digits, even if numerical magnitude and word frequency are controlled. We also observed other effects of the above specific numerical properties, such as multiplication attributes, which facilitated or interfered with the speed of parity judgment. Based on these effects of specific numerical properties we proposed and elaborated a parity continuum account. However, our cross-lingual study also suggests that parity representation and/or access seem to depend on the linguistic properties of the respective language or education and culture. Overall, the results suggest that the “perceived” parity is not the same as objective parity, and some numbers are more prototypical exemplars of their categories.

Parity judgment—that is, deciding whether a number is even or odd—is one of the earliest mathematical tasks learned in school. Formally, parity can take one of two values: an even number is an integer of the form ^{1}

Thus, mathematically, parity is clearly defined. The aim of the present study was, however, to explore how parity is processed cognitively. While cognitive and psychological accounts so far have followed the mathematical definition and defined parity in terms of a categorical psychological representation, the present study aimed at testing an alternative account: Cognitively, parity may be represented in a more gradual manner, such that some numbers are represented as “more odd” or “more even” than other odd or even numbers, respectively. While this may seem an irritating concept for some numerical cognition researchers at first sight, we actually borrow from old ideas, which we apply to the concept of parity. Prototype theory (e.g., Posner and Keele,

Several studies conducted to date have suggested that participants' responses to the parity of different numbers vary. Smallest Space Analyses (SSA-I; Guttman,

Dehaene et al. (^{2}

While some properties are expected to influence the perceived “evenness” of a number, other properties should influence the perceived “oddness,” For instance, whether a number is prime may contribute to its subjective oddness. Notably, numbers 1 and 9 are the only one-digit odd numbers that are not prime numbers, and a reanalysis of data reported by Cipora and Nuerk (

These factors may explain the general patterns in one-digit numbers, but of course cannot be systematically tested in one-digit numbers, given that there are too few numbers and too many degrees of freedom (e.g., almost all one-digit odd numbers are also primes, almost all even one-digit numbers are also powers of two; see above). These confounds are also reflected in the inconclusive results of experiments using single-digit numbers. However, such assumptions can be tested for two-digit numbers, which we therefore set out to investigate here.

We suggest the “parity continuum” as a tentative account of the influence of numerical properties on the parity representation of two-digit numbers. In line with the properties investigated by Dehaene et al. (

Tentative account of numerical properties and perceived parity.

With respect to easiness of division, it is easy to recognize numbers

For even numbers,

Following the easiness of division account, it must be noted that numbers that are

Empirical studies on parity judgments in two-digit numbers indicate that more than just the mathematical properties of the number influence reaction times. Namely, participants tend to respond faster to two-digit numbers if the number's decade and unit have the same parity status (both even: e.g., 48; both odd: e.g., 73), and respond slower if the parity status of the decade and the unit differ from each other (one even, one odd: e.g., 32, 45; Dehaene et al.,

To sum up, properties related to divisibility, sub-base and familiarity as well as parity congruity seem to influence the perceived parity of two-digit numbers. What is more, one can point to a number of linguistic factors that need to be taken into consideration while investigating numerical processing.

Numerical processing is also affected by linguistic features (see e.g., Dowker and Nuerk,

In the case of multi-digit numbers, another linguistic property known as the inversion property is of particular importance. German two-digit number words are inverted: The unit digit is articulated first, followed by the decade digit (e.g., 25 is “fünfundzwanzig”—“five-and-twenty”). In other languages, like English or Polish, the structures of the number word systems are comparable to the Arabic number notation, i.e., the decade digit is articulated first and followed by the unit digit. The inversion property in German can lead to problems with transcoding, i.e., children mixing up units and decades when writing numbers on dictation (Zuber et al.,

However, not only the composition of number words influences number processing, but also the grammatical number (singular, plural) assigned to a number (Roettger and Domahs,

Altogether, linguistic factors are expected to influence number processing, and, therefore, to affect response speed for parity judgment. Thus, we expect reaction times for the examined numerical properties to differ cross-linguistically. Due to these linguistic influences, our initial account might not accurately depict the effect of the odd-even continuum for different language groups.

Numerous studies investigating numerical processing point out that numerical magnitude and frequency of a given number word in natural language affect decision times on numerical stimuli. These effects can be observed both in parity and magnitude judgments. Therefore, we consider them as potentially influencing our results, despite being irrelevant to the postulated parity continuum account.

First of all, processing of numbers is affected by their magnitude. Larger numbers are associated with longer reaction times in number comparison tasks (i.e., the size effect; Moyer and Landauer,

Besides magnitude, the frequency of a number word (Whaley, _{10}) frequency estimates of number words (Gielen et al.,

To sum up, properties such as numerical magnitude and word frequency may play a role for numerical judgments, and thus need to be taken into account, although they are not specifically related to the parity continuum account.

The present study aimed at testing all abovementioned numerical and linguistic factors influencing parity judgments of auditorily presented two-digit numbers within one comprehensive account.

Firstly, according to prototypicality, numbers possessing the properties included in our account (i.e., numbers appearing “more odd”/“more even”) are expected to be associated with shorter reaction times. Alternatively, according to an account based on the markedness strength, as we laid out above, odd numbers are linguistically marked and therefore slower. Linguistically, markedness is a strict category, but psychologically, its effects have been shown to be influenced by individual differences, such as handedness (e.g., Huber et al.,

Secondly, we expected overall between-language differences in parity decisions. Namely, German speakers should show significantly shorter reaction times than the other language groups, since unit-decade inversion leads to the digit relevant for parity judgment (the unit) being pronounced first in German (H2.1). Furthermore, specific features of grammatical number in Polish and English (i.e., grammatical number incongruency in the case of more than half of the numbers in Polish), might possibly lead to slower reaction times in Polish than in English speakers, and also slower than German speakers, both due to inversion property in German and grammatical number incongruencies in Polish (H2.2).

Thirdly, linguistic properties might have specific influences on effects within the parity continuum. Effects related to properties of the decade number should be weaker in German speakers, because they can initiate the response before hearing the decade number. Therefore, they can be less affected by decade magnitude or parity congruity (H3.1). Other specific linguistic differences between the English, Polish, and German language groups are expected to influence the processing of parity (H3.2).

A total of 110 participants (71 female; mean age: 21.8 ± 3.9 years; range: 18–40) took part in the experiment. Out of them, 36 participants were native English speakers (23 female, mean age: 20.2 ± 2.2 years; range: 18–31), 36 were native German speakers (23 female, age: 22.2 ± 3.7 years; range: 18–33) and 38 were native Polish speakers (25 female, mean age: 23.0 ± 4.9 years; range: 18–40). All participants were right-handed and had normal or corrected to normal vision. At the time of testing none of our participants had spent more than 1 year in a foreign linguistic environment. Both parents of all participants were native speakers of the same language. None of the participants suffered from any diagnosed learning, psychiatric or neurological disorder. We obtained approval for testing from the local ethics committees at each site of data collection (York, Tuebingen, and Warsaw). Except for two Polish participants who did not specify their field of study, all participants indicated that they were university students or academic staff at the respective testing sites.

All participants gave their written consent to being tested as a participant in this experiment and were free to withdraw from participation at any point. Participants were compensated with credit points, sweets, or with monetary compensation according to local regulations at testing sites.

The task was a bimanual computerized parity judgment task on two-digit numbers in different notations/modalities (i.e., participants were to decide whether a given number was even or odd), using the “A” (left hand) and “L” (right hand) keys on a keyboard. Response keys were labeled with colored (blue and purple) stickers. The same laptop model was used at each testing site. The task was programmed and data were collected with Presentation 18.1 software (Neurobehavioral Systems Inc., Albany California, USA).

Stimuli were the numbers from 20 to 99 (10–19 in practice sessions). Stimuli were presented as either Arabic numerals, written number words, or auditorily through the computer's speakers. Presentation modality changed after one block and the order of presentation was randomized to avoid order effects. After the first three blocks with different modalities were presented, another three blocks were presented with response-key assignment reversed.

In this article, we decided to focus on results of the auditory presentation, since linguistic effects like unit-decade inversion are expected to be most salient here. It was shown that SNARC/MARC effects can be notation/modality specific (Nuerk et al.,

For the auditory presentation, each trial started with a black fixation square (25 × 25 pixels), which was presented for a random duration between 175 and 250 ms (jittered in steps of 25 ms). Subsequently, a blurred mask was presented on the screen and stimuli were presented through the speakers of the computer until a response was given or for a maximum duration of 3,000 ms. The next trial started after an inter-stimulus-interval (ISI) of 200 ms. During this time, a gray mask covered the screen. The volume of speakers was set to the maximum level, and this corresponded to the natural loudness of a person speaking next to the participant. The numbers were recorded by female native speakers of the respective languages speaking at a regular tempo. The average length of number words differed between languages: in the case of English it was 3.22 syllables, in Polish 4.94 syllables, and in German 4.11 syllables. All recordings were shorter than 1000 ms and were not adjusted to length in order to keep them natural-sounding.

Participants were tested individually. The order of the blocks was counterbalanced across participants. After responding to demographic questions, participants started with the parity judgment task. Both speed and accuracy were stressed in the instructions.

During a break before the change of response-key assignment and after the last block was presented, participants were asked to do paper-pencil tasks that were not further analyzed (LPS-UT3, Kreuzpointner et al.,

Results from practice sessions were not analyzed. The average error rate was 6.34% and errors were not analyzed due to the ceiling effect in a simple task such as parity judgement. Only reaction times associated with correct responses were further analyzed. Due to technical problems, data from three participants (one per language) were not recorded. Reaction times shorter than 200 ms were treated as anticipations and were excluded. Additionally, reaction times that deviated more than ±3 standard deviations from a participant's mean were excluded sequentially with an update of the mean and standard deviation computation after a trial was excluded until no further exclusions occurred (see e.g., Cipora and Nuerk, ^{3}

Within-participant multiple regressions were calculated separately for odd and even numbers. Predictors not specifically related to the parity continuum account were included in both models. These were: (a) Log-transformed (log_{10}) frequency of a number word estimated by subjective ratings, ranging from 0 to 500 (Gielen et al., ^{4}

Binary predictors:

In order to check for predictor collinearity, we calculated correlations between predictors (See Supplementary Material ^{5}

To investigate whether language groups differed in reaction times (H2.1 and H2.2) and regression slopes (H3), respectively, we calculated one-way ANOVAs. In addition, Bayesian ANOVAs were conducted. Posterior probabilities in favor of the null hypothesis model given the data _{0}

To investigate whether the whole sample showed an odd effect (faster mean reaction times for even than for odd numbers in general), a one-way ANOVA was calculated checking for a group difference between even and odd stimuli.

Including all participants, multiple linear regression analysis and subsequent

Predictors influence on overall response times in all three languages.

_{(106)} |
||||
---|---|---|---|---|

Decade magnitude | 8.32 (7.99) | 10.80 | < |
1.04 |

Unit magnitude | 4.24 (5.10) | 8.60 | < |
0.83 |

Parity congruity | −11.40 (23.30) | −5.05 | < |
−0.49 |

Prime number | 24.10 (31.70) | 7.87 | < |
0.76 |

Square | −1.03 (38.10) | −0.28 | 0.779 | −0.03 |

Multiplication table | 7.57 (37.30) | 2.10 | 0.038 | 0.20 |

Divisibility by 5 | 27.70 (39.00) | 7.35 | < |
0.71 |

Frequency | 7.66 (110.00) | 0.73 | 0.468 | 0.07 |

Decade magnitude | 8.29 (8.56) | 10.00 | < |
0.97 |

Unit magnitude | 2.99 (8.13) | 3.80 | < |
0.37 |

Parity congruity | 3.05 (25.10) | 1.25 | 0.213 | 0.12 |

Square | −11.70 (55.40) | −2.18 | −0.21 | |

Multiplication table | 15.40 (39.20) | 4.06 | < |
0.39 |

Power of 2 | 6.36 (49.40) | 1.33 | 0.185 | 0.13 |

Divisibility by 4 | −10.90 (27.10) | −4.17 | < |
−0.40 |

Frequency | 9.35 (240.00) | 0.41 | 0.682 | 0.04 |

Regarding the other predictors,

Subsequently, regression slopes were tested against zero separately for each language group. Checking whether given effects were observed within each language group was a necessary prerequisite for comparing language groups as a next step.

For odd numbers,

Predictors influence on response times separately for each language.

_{(34)} |
_{(34)} |
_{(36)} |
||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Decade magnitude | 9.87 (4.50) | 13.00 | < |
1.25 | −0.18 (5.68) | −0.19 | 0.851 | −0.02 | 14.80 (4.60) | 19.60 | < |
1.89 |

Unit magnitude | 1.14 (3.70) | 1.82 | 0.078 | 0.18 | 5.68 (5.55) | 6.05 | < |
0.59 | 5.72 (4.54) | 7.66 | < |
0.74 |

Parity congruity | −17.00 (18.30) | −5.48 | < |
−0.53 | −5.90 (27.40) | −1.27 | 0.211 | −0.12 | −8.97 (17.70) | −3.08 | −0.30 | |

Prime number | 27.10 (21.10) | 7.58 | < |
0.73 | 1.60 (32.70) | 0.29 | 0.775 | 0.03 | 41.80 (26.70) | 9.52 | < |
0.92 |

Square | −13.20 (35.10) | −2.23 | −0.22 | 3.46 (44.40) | 0.46 | 0.647 | 0.05 | 6.25 (32.10) | 1.18 | 0.245 | 0.11 | |

Multiplication table | 1.04 (31.20) | 0.20 | 0.845 | 0.02 | −15.10 (33.60) | −2.66 | −0.26 | 35.90 (27.80) | 7.87 | < |
0.76 | |

Divisibility by 5 | 38.70 (32.10) | 7.13 | < |
0.69 | −1.12 (34.60) | −0.19 | 0.849 | −0.02 | 44.10 (33.90) | 7.92 | < |
0.77 |

Frequency | 71.20 (90.00) | 4.69 | < |
0.45 | 25.30 (95.00) | 1.57 | 0.127 | 0.15 | −66.30 (91.00) | −4.43 | < |
−0.43 |

_{(34)} |
_{(34)} |
_{(36)} |
||||||||||

Decade magnitude | 9.26 (6.25) | 8.76 | < |
0.85 | 1.10 (7.41) | 0.88 | 0.387 | 0.09 | 14.30 (6.40) | 13.60 | < |
1.32 |

Unit magnitude | 5.13 (5.58) | 5.43 | < |
0.53 | 5.76 (9.77) | 3.49 | 0.34 | −2.08 (6.30) | −2.01 | 0.052 | −0.19 | |

Parity congruity | 3.25 (26.90) | 0.72 | 0.479 | 0.07 | 5.78 (28.60) | 1.19 | 0.241 | 0.12 | 0.79 (20.50) | 0.23 | 0.817 | 0.02 |

Square | −15.20 (51.50) | −1.74 | 0.091 | −0.17 | −32.80 (60.90) | −3.19 | −0.31 | 9.30 (46.50) | 1.22 | 0.231 | 0.12 | |

Multiplication table | 25.10 (33.80) | 4.38 | < |
0.42 | 23.00 (48.00) | 2.83 | 0.27 | −0.21 (29.80) | −0.04 | 0.966 | −0.01 | |

Power of 2 | −5.34 (38.10) | −0.83 | 0.413 | −0.08 | 27.80 (49.20) | 3.34 | 0.32 | −3.04 (53.40) | −0.35 | 0.731 | −0.03 | |

Divisibility by 4 | −29.70 (20.30) | −8.65 | < |
−0.84 | 1.15 (27.80) | 0.25 | 0.807 | 0.02 | −4.36 (22.90) | −1.16 | 0.254 | −0.11 |

Frequency | 212.00 (120.00) | 10.00 | < |
0.97 | −194.00 (210.00) | −5.50 | < |
−0.53 | 4.95 (170.00) | 0.18 | 0.859 | 0.02 |

As regards the other predictors,

For odd numbers, results of

As regards the other predictors,

For odd numbers, being a

As regards the other predictors,

To address _{(2, 214)} = 68.04, _{(1, 214)} = 0.24, ^{6}_{(2, 214)} = 0.02,

Mean reaction times with 95% confidence interval for the English, German, and Polish language group.

For odd numbers, ANOVAs testing for group differences in regression slopes revealed significant differences between language groups for

Predictors influence on response times as compared between three languages.

_{(2, 104)} |
||||||||
---|---|---|---|---|---|---|---|---|

Decade magnitude | 86.41 | < |
0.62 | 0.000 | 0.000 | 1.000 | Very strong for H1 | All groups differ |

Unit magnitude | 11.56 | < |
0.18 | 0.001 | 0.001 | 0.999 | Very strong for H1 | E differs from G and P |

Parity congruity | 2008.00 | 0.139 | 0.04 | 1.000 | 2276.000 | 0.000 | Very strong for H0 | Not applicable |

Prime number | 21.13 | < |
0.29 | 0.000 | 0.000 | 1.000 | Very strong for H1 | All groups differ |

Square | 2799.00 | 0.065 | 0.05 | 0.999 | 1201.000 | 0.001 | Very strong for H0 | Not applicable |

Multiplication table | 24.96 | < |
0.32 | 0.000 | 0.000 | 1.000 | Very strong for H1 | P differs from G and E |

Divisibility by 5 | 19.46 | < |
0.27 | 0.000 | 0.000 | 1.000 | Very strong for H1 | G differs from E and P |

Frequency | 21.73 | < |
0.30 | 0.000 | 0.000 | 1.000 | Very strong for H1 | P differs from G and E |

Decade magnitude | 35.03 | < |
0.40 | 0.000 | 0.000 | 1.000 | Very strong for H1 | All groups differ |

Unit magnitude | 11.04 | < |
0.18 | 0.002 | 0.002 | 0.998 | Very Strong For H1 | P differs from G and E |

Parity congruity | 0.43 | 0.652 | 0.01 | 1.000 | 8113.000 | 0.000 | Very strong for H0 | not applicable |

Square | 6506.00 | 0.11 | 0.059 | 0.063 | 0.941 | Positive for H1 | G differs from E and P | |

Multiplication table | 5363.00 | 0.09 | 0.133 | 0.154 | 0.867 | Positive for H1 | P differs from G and E | |

Power of 2 | 5321.00 | 0.09 | 0.139 | 0.162 | 0.861 | Positive for H1 | G differs from E and P | |

Divisibility by 4 | 16.60 | < |
0.24 | 0.000 | 0.000 | 1.000 | Very strong for H1 | E differs from G and P |

Frequency | 50.33 | < |
0.49 | 0.000 | 0.000 | 1.000 | Very strong for H1 | All groups differ |

Results of a parity judgment task with two-digit numbers in three language groups (English, German, and Polish) were analyzed regarding numerical properties for odd and even numbers in order to verify the parity continuum account and language differences in parity processing. We observed robust language differences in overall reaction times thus confirming hypotheses H2.1 and H2.2. Hypotheses regarding direction of mean slopes (H1), as well as linguistic differences regarding mean slopes (H3) could partially be confirmed and were partially contradicted, which will be discussed below. It was not straightforward to test the tentative account directly, because the postulated categories are neither fully independent of each other nor fully nested (e.g., odd squares are neither a subset of numbers divisible by 5, nor is it the other way around). Instead, after controlling for the effects of

The fundamental assumption that time needed for parity judgments differs considerably depending on numerical properties was confirmed by the data. However, the strict order postulated by neither by the prototypicality nor the markedness strength account was not fully captured.

For odd numbers, being a

This surprising result suggests that different factors might play a role in parity decisions and thus the account considering one dimension only (i.e., easiness of division) seems too simple to explain all numerical influences. Being

Mean slopes with 95% confidence intervals for numerical properties of

In the case of even numbers, being

Our hypotheses regarding differences in mean overall reaction time between language groups were confirmed: German speaking participants reacted the fastest, while Polish speaking participants the slowest (H2.1 and H2.2). In the case of German participants, reaction times were shortest mostly due to the inversion property—the decisive unit number was heard first so that participants could start to give the response, or at least prepare it. This effect was indeed observed and reaction times were the fastest in German participants, despite the considerably larger syllable length of number words in German than in English. On the other hand, Polish speakers were the slowest, which might be either due to the fact that Polish number words were longest, or due to specific grammatical number properties. Note that the point in time at which specific number words are recognized differs across languages. For instance, to accurately categorize the number 91 in Polish, the decisive syllable “je,” being the first syllable of the number of units, appears in the fifth position of the number word “dziewiećdziesiat jeden,” whereas in German the decisive “ein” syllable appears in the first position of the number word “einundneunzig.”

Furthermore, due to the inversion property, one might also expect that numerical properties will affect German speakers to a lesser extent than English and Polish speakers. Interestingly, this was true only in the case of odd numbers. In the case of even numbers, German speakers were highly affected by numerical properties, but Polish speakers were not (cf. Figure

Mean slopes with 95% confidence intervals for numerical properties of

The overall effects of being

Interestingly, for odd numbers being

English, in which no effects were found, may be a mix between Poland and Germany as regards markedness and prototypicality effects. However, we wish to note that the direction of the effect in Polish might be due to suppression. Finally, the effect of

In the case of even numbers, none of the numerical predictors reached significance in Polish speakers. In the case of English and German speakers, the effect of being

Overall, while some language effects pointed in the hypothesized direction, others were pointing in the opposite direction. Possible causes are linguistic, educational, and cultural differences, different saliencies of the prototype and markedness strength hypotheses in different languages, but also methodological issues like a small number of stimuli in some categories and possible collinearities.

To begin with, in the introduction, we outlined the prototype and the markedness strength hypotheses. For even numbers, these hypotheses predicted the same things. Multiplication attributes should lead to faster RT. For odd numbers, they predicted opposite patterns. While the prototype account predicted faster RTs for more prototypical odd numbers (e.g., prime numbers), the markedness strength account predicted longer RTs for such numbers, because they are psychologically more marked and therefore processed even slower.

The predictions for even numbers (

The predictions for odd numbers are more complicated than we had anticipated. Some of the results seem to favor the prototype hypothesis, while other seem to favor the markedness strength hypothesis. Our presumption is that both hypotheses may be valid and that their saliency depends on linguistic, educational and cultural properties. For instance, being a

The factor

The decade magnitude effect was present in both odd and even numbers at the whole sample level as well as in English and Polish, but not in German speakers. Again, this might be due to the inversion property of German.

The results regarding the unit magnitude are also fairly straightforward. It was apparent for both odd and even numbers at the whole sample level. Interestingly, it was present in German speakers for both odd and even numbers, which shows that magnitude effects are present in this language group but are further modulated by linguistic properties for both unit and decade digits in the expected direction. Nevertheless, the effect of unit magnitude was not present for odd numbers in English speakers or even numbers in Polish speakers. Again, the processing of unit magnitude begins later in English and Polish (because there is no inversion) and it might be weaker for the less salient odd numbers than for the more salient even numbers. In sum, the findings for decade and unit magnitude effects for different languages and for different parities largely mimic those observed for the parity congruity effect. Generally, the influence of the unit is larger in German (because of inversion), while the influence of the decade is larger in English and Polish. If there are further differences between parities, magnitude is more likely activated for even parities than for odd parities.

The

In the case of odd numbers in Polish and even numbers in German the effect was in line with predictions, so that higher frequency was associated with shorter reaction times. The effect was not present for odd numbers in German or even numbers in Polish. At the current stage, we do not have an explanation for this interaction between language and parity with regard to frequency effects.

To begin with the hypotheses regarding, linguistic differences were robustly reflected in our results. First of all, German speakers were less affected by decade magnitude than English and Polish speakers. However, the effect of decade magnitude was not totally eliminated in this group. Namely, this group revealed some effects which depended on decade magnitude, such as responding faster to odd numbers that were part of a multiplication table. Such effects can be only explained by the decade number being at least partly processed, because such information can be extracted only when overall numerical magnitude is processed. On the other hand, inconsistent grammatical number did not play a robust role in parity decisions in Polish speakers. This might be due to the fact that numerical processing was not framed in any linguistic context in the present experiment—participants were presented with numbers only, not embedded in any additional phrasing.

Effects of multiplicativity and other numerical variables on parity could be observed but were not always consistent. For even numbers, being a ^{*}7) would be more even than 46. However, this relation is more tentative than for being a

For odd numbers, the interpretation is more difficult, because the prototype and markedness account predict opposing response patterns and our cross-lingual analysis suggest that both may play a role. In line with the outlined markedness strength account, for odd numbers we observed a gradual decrease in response time, starting from prime numbers to numbers that are part of a multiplication table and finally squares. So, 23 (being a prime number) was slower than 27 (being part of the multiplication table (3^{*}9), which was slower than a square number (25, but see below). In contrast to those multiplicativity attributes, divisibility by 5 rather followed the prototypicality, as it slowed down responses: (e.g., 45 was slower than 47 or 49, when all other factors (prime, square number) were partialled out)—this is in line with the idea that numbers divisible by 5 are not typical odd numbers and are therefore slower to be categorized as odd. In sum, for odd numbers, we can say that multiplication attributes influence parity decisions strongly and significantly. However, it seems that we are looking at two opposing effects here, markedness strength and prototypicality, which compete with each other. Therefore, a simple order according to RT like for even numbers cannot be provided so easily.

All in all, however, the current data suggest that not all numbers are equally odd or equally even. Several aspects of two-digit numbers, their multiplicativity, their parity congruity, and in some languages their frequency influence parity categorization. Dependent on language, culture, education and predictor, sometimes less prototypical numbers of a category are slower responded to, corroborating the prototypicality account, while in other cases more marked numbers (and in the case of odd numbers, therefore more prototypical numbers) are slower responded to. Which account is most salient for which language and which attribute is an endeavor for future research. However, we wish to acknowledge that methodological constraints like collinearities or having few members of a category might also have influenced the results and produced suppression and interaction effects. This is not a fault of the current study, as we used all two-digit numbers above 19, but instead an inherent attribute of our numerical system. For instance, there are just two even square numbers between 20 and 99, namely 36 and 64 (note that both of them are divisible by four and one of them is also a power of 2). Of course, 2 members in one category is much less than anybody would have liked. Therefore, independent replications of our results are necessary to see how stable the results for a given language will be.^{7}

Nevertheless, although not every single multiplicativity predictor (especially for small stimulus groups and high collinearity) may prevail in a replication, the present results quite clearly show that the parity judgments are not all the same. There are some consistent findings that unit and decade magnitude, parity congruity, but also some attributes like being a prime number or being divisible by 4 influence parity decisions in a fairly consistent way across languages. Therefore, we believe it is fair after this study to conclude that not all even/odd numbers are psychologically equally even or odd, respectively. However, we also have to acknowledge that the mechanisms responsible for making numbers more even or odd in a given language or culture need to be better studied and understood in the future.

The study was approved by the ethic committee of the Medical Faculty of the University of Tuebingen. It got further approval at other data collection sites (University of York, Department of Psychology and University of Warsaw, Department of Psychology).

KC, MS, KL, SG, FD, MH, and H-CN designed the study. LH, M-LS collected the data. LH, KC, M-LS, and MS analyzed the data. LH, KC, and MS wrote the manuscript. All authors read and commented on and corrected the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We would like to thank all participants. This research was funded by a grant from the DFG (NU 265/3-1) to H-CN supporting KC and MS, and from the National Science Center (NCN), Poland (2014/15/G/HS6/04604) to MH supporting KL. KC, MS, and H-CN are further supported by the LEAD Graduate School & Research Network (GSC1028), which is funded within the framework of the Excellence Initiative of the German federal and state governments. We acknowledge support by Deutsche Forschungsgemeinschaft and Open Access Publishing Fund of University of Tuebingen. Finally, we thank our assistants who helped with data collection and language proofreading the manuscript.

The Supplementary Material for this article can be found online at:

^{1}A group theory in mathematics is about understanding algebraic structures known as groups, which consist of a set of elements and an operation. Here, it provides a background and formal framework for a concept of parity.

^{2}On the one hand this result is surprising as number two can be considered as a prototypical even number. On the other hand, it is also a prime number (i.e., it is divisible only by one and by itself). Even more importantly, number 2 is the only even prime number. This property may, at least in some individuals, lead to longer parity decision times to this number.

^{3}It was demonstrated several times that phenomena observed in numerical cognition, such as for example the SNARC effect are highly dependent on the task set (see e.g., Dehaene et al.,

^{4}We consider this database as the standard in the field of numerical cognition and a better proxy of the real frequency when a cross-lingual design is applied.

^{5}Intercorrelations between predictors are considered problematic if they exceed .80. Another value indicating collinearity is the Variance Inflation Factor (VIF), which should not exceed 10 (see e.g., Field et al.,

^{6}Hines et al. (

^{7}Note that pairwise matching is probably impossible, because this study suggests that so many different attributes (decade magnitude, unit magnitude, parity congruity, frequency and different multiplicativity attributes) may influence reaction times. These would need to be controlled for pairwise matching, which is probably impossible.