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# A Probability-Based Algorithm for Evaluating Climbing Difficulty Grades

Quentin Ansel

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## To cite this version:

Quentin Ansel. A Probability-Based Algorithm for Evaluating Climbing Difficulty Grades. Qeios, 2023, 10.32388/UYLI30. hal-04641837

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https://hal.science/hal-04641837v1

Submitted on 10 Jul 2024

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Distributed under a Creative Commons Attribution 4.0 International License
# A probability-based algorithm for evaluating climbing difficulty grades

Quentin Ansel*

Institut UTINAM, CNRS UMR 6213, Université Bourgogne Franche-Comté,
Observatoire des Sciences de l'Univers THETA, 41 bis avenue de l'Observatoire, F-25010 Besançon, France

(Dated: December 21, 2023)

This paper describes a new mathematical model for the estimation of the grade of a climbing route. The calculation is based on the association of several route and boulder sections separated by rests. Contrary to other similar methods, this model introduces a probabilistic approach describing the uncertainty that one can have about the grade and the different feelings that climbers can have on a route grade. Several aspects of the model are commented and studied. A short comparative study of some of the hardest routes in the world is also presented.

## I. INTRODUCTION

Climbing grades are important features of rock climbing since they aim to indicate the difficulty of each route [1]. They are also a piece of important information for climbers since they tell them how affordable a route is. Grades are also an important for professional climbers searching for the most challenging climbs in the world. Different grade systems exist [1], but they are widely known for being both subjective and representing a kind of underlying truth of a route difficulty [2]. The grades of sport and boulder lines take into account in a complex way the size/shape of the holds, their spacing, their orientation, the steepness, the length of the route, and sometimes the clipping positions. All these aspects put constraints on the human ability to climb a rock. Based on their technical skills, their strength, and their flexibility, climbers process these data to estimate the difficulty. This is a complex task for which there is no real recipe (although machine learning techniques have been tried to predict the grade of boulder problems from the data of hold positions [3]). A climber can propose a grade based on their own experience on a given difficulty range, but the process is at the end, based on personal feeling. The "official" grade of the route is obtained after multiple repetitions of this latter until a consensus is obtained. Mathematically, this can be seen as an average over all the personal grades. This observation leads to the idea that attributing a grade to a route can be modeled mathematically using probability theory [4]. In such a picture, the grade is a random variable that takes its value for each person who climbs the route. As a consequence, a route is attributed to a probability density of grade that tells us how likely a given difficulty level can be felt by someone. The "official" grade is then the expected value.

On another side, there has been an increasing interest in a systematic way of evaluating the grade of a route, from the data of each sequence of this latter. Such an algorithm has existed informally more or less for many years, but the development of a well-defined and effective computation method is very recent [5-7]. Up to date, the most successful method is the one provided by the website *darth-grader.net* [7], called DGC in this paper, for Darth-Grader Calculator. DGC calculates a route grade based on the association of several route and boulder sections separated by rests. In many situations, it returns the correct grade (or at least a likely grade), but it has several drawbacks [8]. (i) It does not take into account the uncertainty or the probabilistic aspects. (ii) It cannot take into account an effect that can be called "a sensitivity to initial conditions" for which a small change in one or several sections can lead to a very different final result. (iii) In some specific situations, it leads to obvious false estimations. For example, a 7B boulder followed without rest with a 6a route (in the French grade system) is rated 7c+ (hard) by DGC, while the reverse route, i.e., a 6a followed without rest by a 7B, returns an 8a (soft) route. The 7B boulder being extremely harder than the 6a route, it must not make any significant difference in the final grade if it comes first or second.

This paper is devoted to the presentation of a mathematical model resolving the issues of DGC listed above. The algorithm is then used for a short comparative study of some of the hardest routes in the world. The program used for the numerical computation is not yet user-friendly nor available on the internet. But these further developments could be of great interest to many practitioners, both amateur and professional.

The structure of the article is the following. First, the mathematical model is presented. The motivations behind the model are explained in detail, and the parameter fitting procedure is discussed (the system depends on a few parameters that must be determined first). Next, the comparative study of a few routes is presented. Finally, a short conclusion is made.

<footnote>* quentin.ansel@univ-fcomte.fr</footnote>
## II. THE MATHEMATICAL MODEL

The model is based on the idea that the grade of a route can be evaluated by decomposing the route into small sections and rests that can be graded individually. The association of all these building blocks must return the grade of the entire line. As will be explained in detail below, the association rule must take into account a few mathematical ingredients. But before going into these details, let us start the discussion with a quite widely believed fact [7], two 7a separated by a medium rest (i.e., a rest which is quite good, but not sufficient to recover completely from the previous physical effort), should be 7b. In mathematical, terms, one would translate this into

$$7a M 7a = 7b \qquad (1)$$

or more generally,

$$g_n M g_n = g_{n+2}. \qquad (2)$$

where $g_n$ denotes the grade $n$ of the grade system, and $M$ denotes a medium rest. The equation (2) is a recurrence relationship. If one knows the difficulty of the first grade levels it is then possible, in principle, to span the entire grade system, to recover all the existing grades, and even the ones which have not been achieved yet. Equation (2) is interesting and easily exploitable for a human, but it is not easily used in a numerical algorithm, and most importantly, it fails to capture nuances. With this relation, it is not possible to model a situation like "these two routes are both 7a, but this one seems quite a bit harder, but not enough for being 7a+".

### A. Energy Associated With a Climbing Grade

To incorporate thin nuances in the estimation of climbing difficulty levels, the mathematical grade system must be defined over real numbers $\mathbb{R}$ and not over integers $\mathbb{N}$. The real number associated with the difficulty of a route is denoted $E$, and it can be interpreted as the energy that must be provided by the climber to send the route. The analogy has, however, some limits since the difficulty does not depend on pure strength, but it also depends on technical abilities, or other physical abilities that reduce the intensity of the effort, such as flexibility. Then, the quantity $E$ is not strictly a physical energy, and it is kept unitless. Another interpretation of the quantity $E$ is one of the ranking points (a little bit like the 8a.nu scores [9]). Achieving a given grade returns a specific number of points. The harder the route, the higher the number of points.

To provide a correspondence between a grade system and the real numbers, it is necessary to fix (in a quite arbitrary way) the energy of the first grades, and then, the reference energy of all the grades can be deduced using equation (2), which becomes $2E_n = E_{n+2}$. Assuming that the first grades are given by 2, 3, and 4a, the energy associated with these grades are assumed to be respectively $E = 1, 2, 3$. Note that for this model, it is chosen to start the grades system with these three grades since it is very difficult to make a distinction between subdivisions (given by letters) in the degrees 2 and 3. The energy associated with each grade from 2 to 10a is given in figure 1. The correspondence between energy and grade as given in this figure is called the energy of reference. We observe an exponential increase of the energy [10] (it is not linear, notice the log scale in the graph). Note that a similar conclusion has been established in Ref. [11], where the authors have performed a rigorous analysis using a Bayesian analysis. A simple curve fitting enables us to determine an approximated formula to estimate the energy of reference of a climbing grade. It is given by

$$E \approx 1.21^{2n} \qquad (3)$$

with $n$ the number corresponding to the grade $g_n$, with $g_1 = 2$, $g_2 = 3$, $g_3 = 4a$, $g_4 = 4b,...$. Of course, the precise values of the energies depend on the initial ones, chosen arbitrarily, but it does not change the fact that the energy increases exponentially with the grade. The exponential increase is a key feature, the precise value of the energy is not as relevant. This observation can be understood by making an analogy with the intensity of sounds [12]. To quantify the strength of a sound, we usually use the decibel unit (dB). This is a log scale and a small difference in dB implies a large difference in energy carried by the sound. This unit system has been chosen because humans are not sensible linearly with the energy received from an acoustic wave. They are only able to distinguish large differences in energy (i.e. if a sound feels a little bit stronger than another one, it does not carry a little bit more energy, it carries a lot more energy). A similar situation happens for the difficulty of climbing routes.
<figure><img src="image_4.png" /><figcaption>FIG. 1. Energy of reference E associated with each climbing grade (French system), from the level 2 to the level 10a. The curve fit as for equation E = 1.21²ⁿ, with n the number associated with the grade gₙ. The correspondence with the first grades is g₁ = 2, g₂ = 3, g₃ = 4a, g₄ = 4b,...</figcaption></figure>

## B. Bouldery vs. Endurance Sections

When sections are short and bouldery, it is harder to attribute a grade similar to the one of an entire route. The common practice is to provide a boulder grade on short hard sections (i.e., ≲ 10 moves) and to keep a route grade for long endurance sections (i.e., ≳ 10 moves). Here again, the French grade (Fontainebleau) system is employed for boulder grades. Despite its similar notation with the route grade system, the difficulty levels and the scale of the boulder system do not correspond to the route system [13]. It was initially assumed a dissimilarity of one degree between boulder grades and route grades. For example, doing a 6A or 6B in Fontainebleau would be as hard as doing a 7a route. This correspondence remains valid for the easiest problems, but it has been modified over the years when new climbing difficulties have been introduced. For example, a 7B boulder corresponds to 8a routes, and a 8B+ boulder corresponds to a 9a route. However, there still exists some discrepancies. The 8B+ ↔ 9a correspondence is now widely assumed for the route Hubble [14], at Raven Tor, but some very long boulders are graded 8C, and they are assumed equivalent to 9a or 9a+ routes (e.g. "la force", at Orsay's roof, "The wheel of life" in the Grampians, "Unendliche Gesichte 1+2+3" in Magic Wood) [15].

To assign an energy value to boulder grades, whose scale is already fixed by the equation (2), a few assumptions are required. In the following, it is assumed that 2 ↔ 2, 6a ↔ 4A, 7a ↔ 6B, 8a ↔ 7B, 9a ↔ 8B+, 9a + /9b ↔ 8C, 9b/9b+ ↔ 8C+, 9b + /9c ↔ 9A. Here, the energy of a slashed grade corresponds to the midpoint energy between the reference energy of the two nearby grades (note that the slash will also be used with another meaning, below in the text). A linear interpolation is used for the grades in between. The last correspondences are based on Adam Ondra's opinion, as given in [13].

## C. Different Association Rules

The next step of the mathematical model is to introduce different association rules of energy. The underlying purpose is to incorporate in the computation the quality of a rest between two sections, and the intensity of the sections (bouldery or endurance). Following the classification of DGC, rests are classified into good (G), medium (M), bad (B), and non-existing (N). A good rest is a rest for which an (almost) complete recovery of the physical abilities is possible. A non-existing rest is by definition the absence of rest, a bad rest is a very short break, where it is possible to shake a bit, and chalk the hands, but it is not sufficient for a complete recovery. A medium rest is something in between a good and a bad rest.

Using the same convention as equation (2), an arbitrary association rule is noted

$$g_n R g_k = g_l \quad (4)$$

with R = G, M, B, N.

The 4 types of rests provide an easy-to-use classification. However, it has some limitations, which are the same as the climbing grades. For the mathematical model, it is preferable to describe a rest with a real number, taken in the interval I = [0, 1]. For simplicity, such a number is also noted R in the following. A rest with R = 0 corresponds to a nonexistent rest, and a rest with R = 1 corresponds to a good rest. Medium and bad rests are defined respectively by R = 0.5 and R = 0.25.
With these materials at hand, it is now possible to translate equation (4) into an operation on energies. It takes the form of a function $E_3 = f(E_1, R, E_2)$, with $f: \mathbb{R} \times I \times \mathbb{R} \rightarrow \mathbb{R}$. Here, the following form of the function is chosen:

$$f(E_1, R, E_2) = E_+ + \beta \left( 1 + \alpha_{1,2}(R) e^{-\frac{(\ln E_2 - \ln E_1)^4}{\sigma^4}} \right) E_- \quad (5)$$

$$E_+ = \max(E_2, E_1) \quad (6)$$

$$E_- = \min(E_2, E_1) \quad (7)$$

$$\alpha_{12}(R) = \begin{cases} R \alpha_{G12} + (1-R)\alpha_{N12} & \text{if } E_1 < E_2 \\ R \alpha_{G21} + (1-R)\alpha_{N21} & \text{if } E_1 > E_2 \\ \frac{R}{2}(\alpha_{G12} + \alpha_{G21}) + \frac{1-R}{2}(\alpha_{N12} + \alpha_{N21}) & \text{if } E_1 = E_2 \end{cases} \quad (8)$$

$$\beta = 1 + \beta_0 e^{-\frac{(\ln E_2 - \ln E_1)^4}{\sigma^4}} \quad (9)$$

where $\beta_0$ is a positive constant different from zero 0 if the second section is a boulder problem. For a long section of endurance, $\beta_0 = 0$. Moreover, $\sigma > 0$ is another parameter to determine. In total, the model has 6 parameters, $\alpha_{G12}, \alpha_{G21}, \alpha_{N12}, \alpha_{N21}, \beta_0, \sigma$. They must be determined from the data of well-established routes. The derivation method of these coefficients is described in section IIF.

The association function may look a little bit complicated, but it possesses the following interesting properties:

* It returns a real number that can be assimilated to the energy of a route.

* It is not commutative, e.g., $8a N 7c$ is not equal to $7c N 8a$.

* It cannot be smaller than the largest energy associated with the two sections (e.g. the energy of $8a G 7a$ cannot be smaller than the energy of $8a$).

* The easiest section contributes significantly to the total grade only if its difficulty level is not too far from the highest one (e.g., the $5b$ in $8a N 5b$ is negligible in front of the $8a$, and the result must be $8a$, but in the case of $8a N 7c$, the $7c$ is not negligible and the final result must be higher than $8a$.)

* The quality of the rest influences the total grade by weighting the contribution of the easiest section. The hardest one remains unaffected by a rest.

* A boulder section coming in second position is usually harder to achieve. Consequently, its energy acquires an added value, but only if the boulder problem resides in the same range of difficulty as the other section.

## D. Introducing probability distributions

In the previous sections, a mathematical model of routes and boulder climbing grades has been developed. An energy quantity has been assigned to any kind of climbing sequence, and association rules have been proposed to compute the energy of a full route. Real numbers have been used to open the possibility that routes or boulders with the same grade can feel more or less hard, but this aspect has not been fully exploited yet. Moreover, the model, as it is presented so far, is not fully satisfactory since it does not well describe the various opinions on the difficulty level of a line. A possible origin for the disparate viewpoints comes from the personal limitations of climbers. For example, a reachy route is harder for small people. This leads to the idea that a grade is not always fully determined, and we have a probability to obtain a given difficulty level. The probability distribution can be spread over different levels (for example 50% $8a+$ and 50% $8b$), but it can also be extremely located on a single one (e.g., 3% $7c+$, 92% $8a$, and 5% $8a+$).

In the mathematical model, the probability distribution is defined by a normalized function $p(E) : \mathbb{R} \rightarrow [0, 1]$. This distribution can be mapped back to the climbing grades using the energy grade correspondence given in figure 1. The switching between two grades $g_n$ and $g_{n+1}$ is defined by the midpoint energy $E_{switching} = \frac{1}{2}(E(g_n) + E(g_{n+1}))$.

With the introduction of probabilities, the energy of a route becomes a random variable. As a consequence, during the computation we have to specify a probability distribution of energy for each climbing section and rest. Many choices of input probability distribution can be decided. For the climbing sections, they are all chosen uniform on the interval $[\ln(E_1), \ln(E_2)]$, where $E_1$ and $E_2$ are respectively the minimum and the maximum energy allowed on the uncertainty interval. A uniform probability distribution on an interval $A$ is noted by $u_A$. The input probability for the energy $E$ is therefore noted $u_{[\ln(E_1), \ln(E_2)]}(E)$. Logarithms are utilized to return approximately a uniform distribution on climbing grades rather than on energy. The values of $E_1$ and $E_2$ are by default the switching energies between
the nearest grades, but, is can also be any energy. The interval of uncertainty can be chosen arbitrarily large. This is useful for sections very difficult to estimate (i.e., the interval can cover more than a single difficulty level).

In addition to the probability distribution on energies, a probability distribution $p(R) : I \to [0, 1]$ is assigned to the rests. They are also assumed uniforms on a subinterval $[R_1, R_2] \in I$, i.e., $p(R) = u_{[R_1, R_2]}(R)$.

Now that input probability distributions are specified, the question is how the probability distribution of a full route can be computed. For each section and rest, random values of energies and rests are generated numerically. With these values, the energy of the full route is computed recursively using the equation (5). The process is repeated many times (typically several thousand repetitions). The final energy of the route is stored in memory for each repetition. Each value of the final energy is different, and they are realizations of a random variable corresponding to the energy of the full route. Then, the probability distribution of the line is reconstructed with the histogram of the computed energies. Note that contrary to the initial probability distributions, the final one is not necessarily uniform.

## E. Summary of the algorithm and illustration through an example

Now that all the ingredients of the mathematical model have been introduced, it is possible to summarize more clearly the calculation steps.

1. Define the probability distribution $u_{[\ln(E_1^{(n)}), \ln(E_2^{(n)})]}$ for each climbing section of the routes, as well as the probability distribution $u_{[R_1^{(n)}, R_2^{(n)}]}$ for each rest of the route. The indices $(n)$ refer to their order of appearance in the full route. The definition of these probabilities consists on specifying $E_1^{(n)}, E_2^{(n)}, R_1^{(n)},$ and $R_2^{(n)}$.

2. For $i = 1$ to $i = N$, with $N$ chosen sufficiently large (quite good values are $N = 2000$ or $N = 5000$), do the following steps:

* Generate random energies $E_i^{(n)}$, and random rest $R_i^{(n)}$, using the probability laws $u_{[\ln(E_1^{(n)}), \ln(E_2^{(n)})]}$ and $u_{[R_1^{(n)}, R_2^{(n)}]}$.

* Compute recursively the energy of the route using $E_i^{(n)}, R_i^{(n)},$ and equation (5). The recursion is performed as follows. First compute $f(E_i^{(1)}, R_i^{(2)}, E_i^{(3)})$, then compute, $f(E_i^{(1)}, R_i^{(2)}, E_i^{(3)}), R_i^{(4)}, E_i^{(5)})$,... until all the sections and rests of the route have been used. The result of this computation corresponds to the final energy of the route, and it is noted $E_{i,final}$.

3. Compute a histogram from the data of the $N$ values $E_{i,final}$. Normalize the histogram to get a probability distribution.

To illustrate the algorithm, the example of an hypothetical route is considered. The line is defined by:

$$g = 7a M 7a M/G 6B/6B + . \qquad (10)$$

The use of slashes differs from the usual convention and its first appearance in this paper. It means that the probability distribution is not centered on a single grade, but the interval used for the probability distribution is determined by the reference energy of the two grades.

The distributions associated with each section of the route are the following:

* For a $7a$ section, $p_{7a}(E) = u_{[\ln 733.5, \ln 1075]}(E)$, with $733.5 = \frac{1}{2}(E_{6c+} + E_{7a})$ and $733.5 = \frac{1}{2}(E_{7a+} + E_{7a})$

* For a medium rest, $p_M(R) = \delta(R - 0.5)$, with $\delta$ the Dirac distribution.

* For a medium/good rest, $p_{M/G}(R) = u_{[0.5,1]}(R)$.

* For a $6B/6B+$ section, $p_{6B/6B+}(E) = u_{[\ln 872, \ln 2166]}(E)$, with $872 = E_{6B}$ and $2166 = E_{6B+}$.

The probability distribution of the full route is computed in two steps, each step being given by the computation of the map $f$. The first application of $f$ returns us a distribution $p_{intermediate} = f(p_{7a}, p_M, p_{7a})$, which corresponds to the distribution associated with $7a M 7a$. The second use of $f$ provide us the final probability distribution $p_{final} = f(p_{intermediate}, p_M, p_{6B/6B+})$. Of course, this last distribution corresponds to equation (10).

The probability distributions $p_{7a}(E)$, $p_{intermediate}(E)$, and $p_{final}(E)$ are plotted in figure 2. Several observations can be made. First, the shape of the distribution is not conserved. The initial distribution is (almost) flat, the small variations are numerical artifacts induced by the finite sampling of the distribution. The second distribution is closer to a normal law, and the third one has two distinct peaks. The accumulation of uncertainties at each section of the
<figure><img src="image_5.png" /><figcaption>FIG. 2. (Left) smooth probability distribution $p_{7a}(E)$, $p_{intermediate}(E)$, and $p_{final}(E)$. To simplify the reading, the abscissa is in a log scale, and reference energy values are replaced by their corresponding grade. (Right) probability $p(g_n)$ to find a given grade for the three cases of the left panel (see equation (11) for its definition).</figcaption></figure>

route is responsible for a large uncertainty in the final difficulty level. For most people, it is likely a 7c route, but 7b+ and 7c+ are also possible. In fact, almost fifty percent of the climbers may disagree on the final grade. In the end, the 7c may remain the "official" grade, but no clear consensus may be achieved. Interestingly, we recover the property illustrated in equation (2). This is a key observation for the consistency of the model. Indeed, equation (2) was used to define the reference energies, but it is not used anymore in the definition of $f$ (equation (5)). It is therefore non-trivial that $f$ conserves the relation (2).

The continuous probability distribution $p(E)$ provides precise information on the difficulty of the line, but it does not directly answer the following question: what is the probability of getting a given grade? In other words, we would like to pass from a continuous description with the variable $E$ to a discrete description with the variable $g_n$, expressed in a usual grade system. The mapping is performed with the integral:

$$p(g_n) = \int_{(E_{n-1}+E_n)/2}^{(E_{n+1}+E_n)/2} p(E) \, dE. \qquad (11)$$

These discrete probabilities are given in the right panel of figure 2.

## F. Computation of the Model Parameters

The computation of the free parameters of the model is a nontrivial task. They can be determined by fitting the model with experimental data, i.e., data of well-established routes.

A collection of 33 well-characterized routes is used as a training dataset for the model (see appendix). The number of elements in the dataset is small compared to typical machine learning problems [16, 17], but here, the number of parameters of the model is small and it does not seem necessary to use datasets with thousands or millions of elements. The dataset entries are chosen to represent as much as possible all the possible kinds of associations. The parameter fitting of $\alpha_{G12}$, $\alpha_{G21}$, $\alpha_{N12}$, $\alpha_{N21}$, and $\beta_0$ is performed by minimizing the cost function:

$$C = \frac{1}{N_{route}} \sum_{r=1}^{N_{route}} |\ln(E_{r, target}) - \ln(\langle E_{r, final} \rangle)| \qquad (12)$$

with, $N_{route} = 33$ the number of routes in the training dataset, $E_{r, target}$ is the target energy associated with the grade of the route $r$ of the dataset. This energy is assumed known without uncertainty, and the grade/energy correspondence is the one given in figure 1. $\langle E_{r, final} \rangle$ denotes the estimated mean value of the final energy of the route $r$, as computed by the algorithm. This energy depends directly on the series of climbing sequences and rests of the line (which are specified in the dataset), and it also depends on the parameters to fit. Note that $C$ can be evaluated with good accuracy using only a few random realizations of $\langle E_{r, final} \rangle$ (order of 50 compared to an order of 5000 to evaluate the probability distribution correctly). The cost function $F$ is then minimized with the algorithm JAYA [18]. The result of the optimization is: $\alpha_{G12} = 0.0016$, $\alpha_{G21} = 0.2873$, $\alpha_{N12} = 0.8485$, $\alpha_{N21} = 0.3828$, and $\beta_0 = 0.3754$.
<figure><img src="image_6.png" /><figcaption>FIG. 3. Probabilities $p(g_n)$ predicted by the algorithm. The target (official) grades of the routes are given after their names in brackets. As we can observe, the probabilities are always highly concentrated around the target grade.</figcaption></figure>

The parameter $\sigma$ is not determined like the other parameters. It is fixed by hand to $\sigma = 20$. The dataset does not allow us to determine it very accurately, but having an accurate value does not seem strictly necessary. It is chosen large enough so that only nearby grades can influence each other, but not too much, to limit the influence of a small grade on a large one.

With this kind of a mathematical model, which is aimed at predicting new data from the basis of known ones (i.e., like in machine-learning techniques), another dataset called the validation set is used to verify the model predictive power. In typical machine learning systems, the training dataset is around 80% of all the data, and the validation dataset is the remaining 20% [16, 17]. Here the validation set is composed of 8 routes with different grades (see appendix for details). The probabilities $p(g_n)$ of these routes, as predicted by the algorithm are given in figure 3. We observe a very good correspondence between the target grades and the highest-grade probability predicted by the algorithm, hence validating the relevance of the model.

### III. APPLICATION: COMPARISON OF SOME OF THE WORLD'S HARDEST ROUTES

To finish this paper, the algorithm is applied in a short comparative analysis of some of the hardest routes in the world (at the moment of writing this paper).

The routes are Silence (9c) [15], DNA (9c) [19], B.I.G. (9c) [20], Sleeping Lion (9b+) [21], and Excalibur (9b+) [22]. The lines have been visited by several climbers, but since their first ascent, none has been repeated yet [23]. The first ascents were made respectively by Adam Ondra (2017), Seb Bouin (2022), Jackob Schubert (2023), Chris Sharma (2023), and Stefano Ghisolfi (2023). For each of these climbers, these routes may be considered as their hardest achievement so far.

The grade density probabilities are computed with the following sequences:

* Silence: $8b G 8C/8C + B/M 8A/8B B 7C + G 7B$,

* DNA: $8c/8c + M/G 8A/8B B 8A + /8B N/B 8c+$,

* B.I.G: $9b N/B 8A/8B N 8a/8a+$,

* Sleeping Lion: $7B/7B + G 8A M/G 8A B/M 8A/8A + N/B 8A$

* Excalibur: $8B B/M 8C$.
It should be noted that some sequences of Silence and DNA are slightly different from the ones initially given after the first ascent, due to additional opinions given by other climbers. For Silence, the first crux may be slightly harder than 8C since it has been tried by at least 3 other climbers, but only A. Ondra has done it yet. Concerning the second crux of the route, it may be easier than initially expected. For DNA, after the comments on J. Schubert, the boulder problems can be harder than the grades given by S. Bouin, especially the first one which has a reachy move.

The probability density $p(E)$ of each route is given in figure III. The grade probabilities $p(g_n)$ are given in table III. We observe that all the routes are predicted in the range $9b+ / 9c$, but only Silence can be considered as a pure $9c$ since it has a 100% probability on this grade. The second hardest route is DNA but with a $9c$ probability around 55%. All the other routes are more likely to be $9b+$. However, several comments on these results can be made.

First of all, Sleeping Lion is very close to DNA, the difficulty seems also clearly above B.I.G. Then, Chris Sharma could have announced $9c$ for this route. The frontier is very close. If we consider that the two middle rests are respectively M and B, the $9c$ probability goes to 69%. If the 8A boulder problems are all replaced by $7C + /8A$ boulders and the $8A/8A+$ boulder is replaced by a 8A boulder, the grade falls to $9b+$ with a probability of 99%.

As a second comment, the density probabilities of B.I.G and Excalibur are quite similar. They are also very large, and this complicates the choice of a grade. For B.I.G, the arguments were clearly asserted by Ondra and Schubert: the route is very long, and the crux feels very hard coming from the ground. Moreover, the route remains more or less humid all the time, and this is a source of additional difficulty. The $9c$ possibility is small, but for these two climbers, the route feels harder than all the other $9b+$ they have made. For Excalibur, this is quite the opposite. The route is short, and so is the duration of the effort. The key to success relies on the middle rest. For a medium rest, the grade is $9b+$ at 72% (still, it remains a hard one), but for a very bad or nonexistent rest (let us say, there is no rest at all), the route is $9c$ at only 60%.

To conclude this small application of the algorithm, it enables us to render quite accurately the true difficulty of a route. A clear consensus on the grade of Silence may be easily achieved, but this may not be the case for DNA, B.I.G., and Sleeping Lion. Their grade could be adjusted by the next repeaters. With these lines very difficult to grade, the maths do not always return the final conclusion. One also has to answer the question, "Which grade do you want for this route". As A. Ondra said about B.I.G [24]: "I think this amazing route deserves a nice grade, too". This is not the first time that such a thing has happened! Maybe almost all the climbing crags have amazing routes with nice grades?

<figure><img src="image_7.png" /><figcaption>FIG. 4. Probability $p(E)$ to obtain a given grade for DNA, Silence, B.I.G., Sleeping Lion, and Excalibur. To simplify the reading, the abscissa is in a log scale, and reference energy values are replaced by their corresponding grade.</figcaption></figure>

<table><caption>TABLE I. Probability $p(g_n)$ to obtain a given grade for DNA, Silence, B.I.G., Sleeping Lion, and Excalibur.</caption><thead><tr><th></th><th>9b+</th><th>9c</th></tr></thead><tbody><tr><td>DNA</td><td>45%</td><td>55%</td></tr><tr><td>Silence</td><td>0%</td><td>100%</td></tr><tr><td>B.I.G.</td><td>84%</td><td>16%</td></tr><tr><td>Sleeping Lion</td><td>77%</td><td>23%</td></tr><tr><td>Excalibur</td><td>65%</td><td>35%</td></tr></tbody></table>
IV. CONCLUSION

In this paper, a mathematical model for the estimation of climbing route grades was presented. The calculation is based on the association of several route and boulder sections separated by rests. The model incorporates several key ingredients. First, the level of difficulty and the quality of a rest in quantified using real numbers. They are interpreted respectively by the amount of energy that must be provided to achieve the route (or in some sense the number of rewarding points for sending a given difficulty), and a percentage of the quality of the rest. Based on real-life observations, a correspondence grade/energy and "type of rest"/"percentage of rest" is made. Interestingly, the energy scales exponentially with the grade. A second important feature is the association rule that allows us to combine the energy of sections separated by rests into the energy associated with the full line. Using the grade/energy correspondence, the grade of the full section can be deduced. The association rule depends on a few parameters derived by a fitting procedure. This latter is performed utilizing an optimization algorithm that computes the best set of parameters reproducing the difficulty level of a dataset of well-characterized routes. The fitted model is then checked using a second dataset. The last important point is the incorporation of probability densities in the model. The use of probabilities is motivated by the various difficulty level that climbers may witness on the same line. In this setting, the grade attributed by each climber is a random variable, and the "official" grade, the grade given in a guidebook for instance, is the expected value.

As an illustrative example, the algorithm was applied to compare a few of the most difficult routes of the world: Silence, DNA, B.I.G., Sleeping Lion, and Excalibur. This short comparative study emphasizes very different situations: cases where there is no possible uncertainty of the grade (everybody will agree on it), and cases where a consensus is far more difficult. Several motivations used to choose a grade rather than another have also been reviewed.

This mathematical model can be further improved to take into account additional features. For example, the addition of very physical sections may not have the same impact as the addition of more technical sections. Moreover, the fitting parameters may also be improved with a bigger dataset. Another possibility is to replace the equation (5) with a neural network. A neural network usually has a very large number of free parameters that must be fixed with the training procedure. This large number of degrees of freedom in the model can enable us to reduce many assumptions (such as the form of eq. (5)). Basically, it would learn the association rule from nothing else than the dataset. This may be very interesting, but it would require a very huge dataset and heavy numerical computations for the training.

As a final remark, the algorithm can also be used to predict the grade of boulder problems if the moves or the sections are graded individually. This kind of computation is straightforward, it is only sufficient to replace the route energies of reference by the boulder energies of references in the computation of the histogram. As an example, the algorithm predict a 100% probability on 9A for *Burden of Dreams*, which is the correct grade according to community (for this computation, the following sequence is used: 8B + N 7C N 7C + N 7A/B 8A).

Appendix A: Datasets

The dataset used for the parameter fitting is given in the table II. The dataset used to validate the fitted parameters is given in the table III.
<table><caption>TABLE II. Dataset used for the parameter fitting.</caption><thead><tr><th>Route Name</th><th>Location</th><th>Target Grade</th><th>Sequences</th></tr></thead><tbody><tr><td>Silence</td><td>Flatenger</td><td>9c</td><td>8b G 8C/C+ B/M 8A+/8B B 7C+ G 7B</td></tr><tr><td>Change</td><td>Flatenger</td><td>9b+</td><td>7C M/G 8B+/ G 7C N 7a/b G 9a</td></tr><tr><td>Bibliographie</td><td>Céüse</td><td>9b+</td><td>8b+ G 8A+/B B 9a</td></tr><tr><td>La Dura Dura</td><td>Oliana</td><td>9b+</td><td>7C N 7C N 8B+ M 8c/c+ M 8a</td></tr><tr><td>Zvěřniec</td><td>Holštejn</td><td>9b+</td><td>9a+ G 8B+ N/B 7A</td></tr><tr><td>Vasil Vasil</td><td>Sloup</td><td>9b+</td><td>8b N 8B+ N 7A</td></tr><tr><td>Taurus</td><td>Byci Skala</td><td>9b</td><td>8C+ N 8b</td></tr><tr><td>Chaxi Raxi</td><td>Oliana</td><td>9b</td><td>8B+ G 9a/a+</td></tr><tr><td>Move</td><td>Flatenger</td><td>9b+</td><td>9a B/G 8B+</td></tr><tr><td>Beyond Integral</td><td>Pic St. loup</td><td>9b</td><td>9a+ G 8A+</td></tr><tr><td>Biographie</td><td>Céüse</td><td>9a+</td><td>8c+ M 7C+/8A M 7c</td></tr><tr><td>Papichulo</td><td>Oliana</td><td>9a+</td><td>8b/b+ G 8c/c+ B 8b+/c</td></tr><tr><td>Les Yeux Plus Gros que l'Antre</td><td>Russan</td><td>9a+</td><td>8c G 9a</td></tr><tr><td>Pornographie (with kneepad)</td><td>Céüse</td><td>8c+</td><td>8c B 8b</td></tr><tr><td>Mange ta Soupe</td><td>St. Pancrasse</td><td>8c</td><td>8a+ N 7B+</td></tr><tr><td>Le minimum</td><td>Buoux</td><td>8c</td><td>7c N 7C M 7c</td></tr><tr><td>Eliot Le Pilote</td><td>St. Ange</td><td>8b+</td><td>8a+ G 7B</td></tr><tr><td>Check Up Gros</td><td>Parmelan</td><td>8b+</td><td>7B+ B 8a+</td></tr><tr><td>La Rage de Vivre</td><td>buoux</td><td>8b+</td><td>8b G 8a+</td></tr><tr><td>Clafoutis</td><td>Mouxy</td><td>8a+</td><td>7c+ M 7c+</td></tr><tr><td>Mo..Mo..Motus</td><td>La Baume du Syratu</td><td>8a</td><td>7b+ G 7c+</td></tr><tr><td>Gras Mouillé</td><td>Romeyer</td><td>8a</td><td>7c M 7c</td></tr><tr><td>Appel au Roi</td><td>Barmaud</td><td>7c+</td><td>6C N 7a+/b</td></tr><tr><td>Les samares</td><td>Barmaud</td><td>7c</td><td>7a/a+ G 6B G 6B G 6c</td></tr><tr><td>Écureuil Plongeur</td><td>Barmaud</td><td>7c</td><td>7b M 7b</td></tr><tr><td>La Chose</td><td>Céüse</td><td>7c</td><td>6C+ G 7a</td></tr><tr><td>Télémaque Contrôle</td><td>Saint Léger</td><td>7b+</td><td>7a+/b G 7a+</td></tr><tr><td>Galaxy</td><td>Céüse</td><td>7b+</td><td>6C G 7a+</td></tr><tr><td>Le Concasseur</td><td>Tamée</td><td>7b+</td><td>6c G 6B N 7a</td></tr><tr><td>Miaou</td><td>Mouxy</td><td>7b</td><td>6c+ B 7a</td></tr><tr><td>Une Ténébreuse Affaire</td><td>Barmaud</td><td>7a</td><td>6c G 6c</td></tr><tr><td>Œstrogène</td><td>La Brême</td><td>6b+</td><td>6b G 6a+</td></tr><tr><td>Les Aigles</td><td>La brême</td><td>5c</td><td>5a+ G 5b+</td></tr></tbody></table>

<table><caption>TABLE III. Dataset used to validate the fitted parameters.</caption><thead><tr><th>Route Name</th><th>Location</th><th>Target Grade</th><th>Sequences</th></tr></thead><tbody><tr><td>First Round First Minute</td><td>Margalef</td><td>9b</td><td>7C+ N/B 8A+ N/B 7B+/C</td></tr><tr><td>Qui</td><td>Geisterschmiedwand</td><td>9a+</td><td>8c M 8A+</td></tr><tr><td>Joe Blau</td><td>Oliana</td><td>8c+</td><td>8a+ G 8b+ M 8b</td></tr><tr><td>Le Nabab</td><td>Saint Léger</td><td>8b+</td><td>8a M 8a M 8a G 7b+</td></tr><tr><td>Mort aux voleurs de feux</td><td>Le Quint</td><td>8a+</td><td>6a+ N 7B G 6B</td></tr><tr><td>Le Grand Pouvoir</td><td>Le Quint</td><td>8a</td><td>7b N/B 6C/C+</td></tr><tr><td>Coup de Pied de Biche</td><td>Rurey</td><td>7c</td><td>6B+/C G 7b</td></tr><tr><td>Est pas P'tit Pois Moi</td><td>Rurey</td><td>7b</td><td>6a G 7a+ M 6c</td></tr></tbody></table>

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[7] DARTH GRADER - the grade calculator ().
[8] At the moment of writing this paper, October 2023.

[9] Ranking • sportclimbing - 8a.nu ().

[10] This is a key point that differs significantly from the approach of DGC. In the technical notes of the website, it is written that grades follow a linear relation. This interpretation is wrong. Relation (2) does not lead to something linear. However, the calculation method of the website is correct in the sense that relation (2) is obtained by the calculator.

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[19] Grimper, Seb bouin enchaîne DNA, possible second 9c de l'histoire...

[20] Jakob schubert (@jakob.schubert) • photos et vidéos instagram ().

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[22] P. Délas, Stefano ghisolfi libère excalibur 9b+ ! - stefano ghisolfi frees excalibur 9b+ !

[23] 8a.nu: Global climbing news ().

[24] B. \.. M. www.benes michl.cz, My reaction on grading b.i.g. 9c by jakob schubert.