Guide number (English Wikipedia)

Jacobson, Ralph (2000). Manual of Photography (9th ed.). Focal Press. p. 331. ISBN 978-0-240-51574-8.
Mortensen, William (1941). Flash in Modern Photography. Camera Craft Publishing Co. p. 47.

bhphotovideo.com

B & H Foto & Electronics: Understanding Guide Numbers
One example of using the single-prime (foot symbol) when marketing in the U.S. is as practiced by Bolt, which specifies the guide number of their VS 510P (product page here) as follows:
The VS-510P Wireless TTL Shoe Mount Flash from Bolt is a dedicated TTL flash for use with Pentax & Samsung DSLR, mirrorless or point-and-shoot cameras. The VS-510 features a guide number of 141' at ISO 100 and an adjustable tilt head with five positions: 0 to 90°.
Another notable example of how flash device guide numbers are communicated to the U.S. market is the large retailer, B & H Foto & Electronics (on-camera flash page here), which states guide numbers formatted like this example for Nissin's i60A:
• Guide Number: 197' at ISO 100 and 200mm

boltflashes.com

One example of using the single-prime (foot symbol) when marketing in the U.S. is as practiced by Bolt, which specifies the guide number of their VS 510P (product page here) as follows:
The VS-510P Wireless TTL Shoe Mount Flash from Bolt is a dedicated TTL flash for use with Pentax & Samsung DSLR, mirrorless or point-and-shoot cameras. The VS-510 features a guide number of 141' at ISO 100 and an adjustable tilt head with five positions: 0 to 90°.
Another notable example of how flash device guide numbers are communicated to the U.S. market is the large retailer, B & H Foto & Electronics (on-camera flash page here), which states guide numbers formatted like this example for Nissin's i60A:
• Guide Number: 197' at ISO 100 and 200mm

canon.com

usa.canon.com

One notable way of expressing guide numbers scaled for use with both feet and meters (and relative to an ISO setting of 100) is as practiced by Metz mecatech GmbH, as exemplified by the data sheet for their mecablitz 52 AF-1 digital (product page here):
Flash output
• High max. guide number 52 (meters), High max. guide
number 170 (feet) for ISO 100/21° and 105 mm
Another notable example is Nikon's practice for expressing the technical specifications of products marketed in the U.S., such as their SB-5000 AF Speedlight (product page here):
Guide Number
34.5 m/113 ft. (at 35 mm) 55
m/180 ft. (at 200 mm) (FX
format, standard illumination
pattern) (at ISO 100)
Yet another notable example is Canon USA's practices, as exemplified by their Speedlite 430EX III-RT (product page here), which is as follows:
Guide Number The maximum Guide No. is approximately 141.1 ft./43m at ISO 100 and 105 mm flash coverage.

dougkerr.net

Whereas scene reflectance (the albedo of the objects being photographed) have no influence on the illuminance (lux value) arriving at any given scene, reflective surfaces along the path between a flash device and a scene, such as light-colored ceilings, do have an influence. Guide numbers are often calculated by manufacturers for optimum results in average indoor settings. See Characterizing the Output of Photographic Flash Units, by Douglas A. Kerr (PDF here) for more.
The continuous-light portion of fill-flash photos
The continuous-light exposures underlying the above tables are in accordance with Eastman Kodak Company's guidelines for exposing film. For example, as exemplified by the technical data sheet for their Kodak Professional Ektar 100 film (PDF here), scenes that are lit by bright or hazy sun with distinct shadows and which are shot between 2 hours after sunrise and 2 hours before sunset are an illuminance of 40,960 lux Archived 2017-08-14 at the Wayback Machine (EV 14 @ ISO 100). For this illuminance, Kodak recommends settings (ISO 100, f/11, ¹/₁₂₅ th of a second) that equal a luminous exposure of 328 lux⋅seconds. This exposure level applies proportionally across the full gamut of Kodak's color and panchromatic B&W films where—at 40,960 lux and f/11—a film speed of ISO 200 is properly exposed when a scene receives a luminous exposure of 164 lux⋅seconds, ISO 400 requires 82 lux⋅seconds, and so forth. Kodak's guidance for a proper continuous-light exposure for all film speeds, scene illuminance, and camera settings reduces as follows:

${\frac {E_{v}\cdot t\cdot S}{f^{2}}}=k$

…where,

E_v is scene illuminance in lux (e.g. 40,960; 81,900; and 111,000). A table of lux values vs. EV @ ISO 100 is available here, by Sekonic. Archived 2017-08-14 at the Wayback Machine Archived 2017-08-14 at the Wayback Machine

t is exposure time in seconds (e.g. 0.008 for ¹/₁₂₅ th of a second)

S is ISO arithmetic speed or sensitivity (e.g. 100, 200, and 400)

f is the f-number (e.g. 8 for an f‑stop of f/8. Common f‑stops from the ${\textstyle {\sqrt {2}}^{\ n}}$ series include f/1.1892, f/1.4142, f/2, f/2.8284, f/4, f/5.6569, f/8, f/11.3137, f/16, f/22.6274, and f/32)

k is a constant equaling 256

When utilizing fill flash, where balancing flash and continuous light can be difficult, the following four derivatives of this continuous-light exposure equation can be useful:

${\sqrt {\frac {E_{v}\cdot t\cdot S}{k}}}=f$

${\frac {{f^{2}}\cdot {k}}{E_{v}\cdot t}}=S$

${\frac {{f^{2}}\cdot {k}}{t\cdot S}}=E_{v}$

${\frac {{f^{2}}\cdot {k}}{E_{v}\cdot S}}=t$ (take ⁠1/1⁠ to obtain the denominator of the fractional shutter speed)

For any combination of lighting, film, and camera settings that conforms to one of the above five equations, a proper luminous exposure is calculated as follows:

$E_{v}\cdot t=H_{v}$

…where,

E_v is scene illuminance in lux (e.g. 40,960; 81,900; and 111,000)

t is exposure time in seconds (e.g. 0.008 for ¹/₁₂₅ th of a second)

H_v is luminous exposure in lux⋅seconds

Note that Kodak's exposure guidelines—for photographs taken in typical settings without the benefit of incident-light meters—are for pictures shot during a broad portion of the day with even some light haze in the sky; this is half as bright as the clear-sky, near-noon, open-area, "sunny f/16 rule", which is EV 15 at ISO 100, or 81,900 lux. Notwithstanding differences in the assumed daylight illuminance, a proper continuous-light exposure is still governed by the same mathematical relationship where (E_v⋅t⋅S)/f² = k.
Flash exposures vs. continuous-light exposures
Importantly, Eastman Kodak Company's exposure recommendations for electronic flash are slightly different from those for continuous-light exposures. Kodak calculates exposures for electronic flash devices according to the following formula:

$GN={\sqrt {\frac {BCPS\times ISO}{20}}}$

…where,

GN is guide number

BCPS is beam candlepower seconds

ISO is ISO arithmetic speed or sensitivity (e.g. 100, 200, and 400)

See Characterizing the Output of Photographic Flash Units, by Douglas A. Kerr (PDF here) for more.

This formula means that under the same conditions used above for Kodak Professional Ektar 100 film (ISO 100 and f/11), a proper electronic flash exposure equals only 270 lux⋅seconds (one candlepower equals 0.981 candela, which equals 0.981 lux⋅second at a distance of one meter). This slightly reduced flash exposure (versus the continuous-light value of 328 lux⋅seconds) compensates for reflections off light colored ceilings, which is typical for indoor flash photography.
Outdoors mixing of continuous-light and GN-based fill flash
This all means that in outdoor settings, flash devices must be roughly nine-percent closer ${\textstyle ({\sqrt {270/328}}=90.7\%)}$ than their indoor-rated guide numbers indicate in order to obtain a full-rated luminous exposure. The above fill-flash tables give distances that have been adjusted accordingly.
Also bear in mind that the distances given in the above tables do not obey the f‑number × distance formula when using the f‑stops shown in the first column because the fill-flash contribution is one f‑stop less than the sunlit portions of the scene—a 50% underexposure. The values may be correctly calculated by hand when the apertures shown in the first column are larger (numerically smaller f‑number) by one f‑stop. Those larger apertures, when expressed in industry-standard nomenclature, are f/11, f/8, f/6.3 (f/5.6 +⅓), and f/5.6 but have the following more precise values underlying the above tables: f/11.3137, f/8, f/6.3496, and f/5.6569 ${\textstyle {({\sqrt {2}}}^{\ 7}}$ , ${\textstyle {\sqrt {2}}^{\ 6}}$ , ${\textstyle {\sqrt {2}}^{\ 5{\tfrac {1}{3}}}}$ , and ${\textstyle {\sqrt {2}}^{\ 5})}$ .

flashbulbs.com

Flashbulbs.com: "Flash info"

kodakalaris.com

imaging.kodakalaris.com

The continuous-light portion of fill-flash photos
The continuous-light exposures underlying the above tables are in accordance with Eastman Kodak Company's guidelines for exposing film. For example, as exemplified by the technical data sheet for their Kodak Professional Ektar 100 film (PDF here), scenes that are lit by bright or hazy sun with distinct shadows and which are shot between 2 hours after sunrise and 2 hours before sunset are an illuminance of 40,960 lux Archived 2017-08-14 at the Wayback Machine (EV 14 @ ISO 100). For this illuminance, Kodak recommends settings (ISO 100, f/11, ¹/₁₂₅ th of a second) that equal a luminous exposure of 328 lux⋅seconds. This exposure level applies proportionally across the full gamut of Kodak's color and panchromatic B&W films where—at 40,960 lux and f/11—a film speed of ISO 200 is properly exposed when a scene receives a luminous exposure of 164 lux⋅seconds, ISO 400 requires 82 lux⋅seconds, and so forth. Kodak's guidance for a proper continuous-light exposure for all film speeds, scene illuminance, and camera settings reduces as follows:

${\frac {E_{v}\cdot t\cdot S}{f^{2}}}=k$

…where,

E_v is scene illuminance in lux (e.g. 40,960; 81,900; and 111,000). A table of lux values vs. EV @ ISO 100 is available here, by Sekonic. Archived 2017-08-14 at the Wayback Machine Archived 2017-08-14 at the Wayback Machine

t is exposure time in seconds (e.g. 0.008 for ¹/₁₂₅ th of a second)

S is ISO arithmetic speed or sensitivity (e.g. 100, 200, and 400)

f is the f-number (e.g. 8 for an f‑stop of f/8. Common f‑stops from the ${\textstyle {\sqrt {2}}^{\ n}}$ series include f/1.1892, f/1.4142, f/2, f/2.8284, f/4, f/5.6569, f/8, f/11.3137, f/16, f/22.6274, and f/32)

k is a constant equaling 256

When utilizing fill flash, where balancing flash and continuous light can be difficult, the following four derivatives of this continuous-light exposure equation can be useful:

${\sqrt {\frac {E_{v}\cdot t\cdot S}{k}}}=f$

${\frac {{f^{2}}\cdot {k}}{E_{v}\cdot t}}=S$

${\frac {{f^{2}}\cdot {k}}{t\cdot S}}=E_{v}$

${\frac {{f^{2}}\cdot {k}}{E_{v}\cdot S}}=t$ (take ⁠1/1⁠ to obtain the denominator of the fractional shutter speed)

For any combination of lighting, film, and camera settings that conforms to one of the above five equations, a proper luminous exposure is calculated as follows:

$E_{v}\cdot t=H_{v}$

…where,

E_v is scene illuminance in lux (e.g. 40,960; 81,900; and 111,000)

t is exposure time in seconds (e.g. 0.008 for ¹/₁₂₅ th of a second)

H_v is luminous exposure in lux⋅seconds

Note that Kodak's exposure guidelines—for photographs taken in typical settings without the benefit of incident-light meters—are for pictures shot during a broad portion of the day with even some light haze in the sky; this is half as bright as the clear-sky, near-noon, open-area, "sunny f/16 rule", which is EV 15 at ISO 100, or 81,900 lux. Notwithstanding differences in the assumed daylight illuminance, a proper continuous-light exposure is still governed by the same mathematical relationship where (E_v⋅t⋅S)/f² = k.
Flash exposures vs. continuous-light exposures
Importantly, Eastman Kodak Company's exposure recommendations for electronic flash are slightly different from those for continuous-light exposures. Kodak calculates exposures for electronic flash devices according to the following formula:

$GN={\sqrt {\frac {BCPS\times ISO}{20}}}$

…where,

GN is guide number

BCPS is beam candlepower seconds

ISO is ISO arithmetic speed or sensitivity (e.g. 100, 200, and 400)

See Characterizing the Output of Photographic Flash Units, by Douglas A. Kerr (PDF here) for more.

This formula means that under the same conditions used above for Kodak Professional Ektar 100 film (ISO 100 and f/11), a proper electronic flash exposure equals only 270 lux⋅seconds (one candlepower equals 0.981 candela, which equals 0.981 lux⋅second at a distance of one meter). This slightly reduced flash exposure (versus the continuous-light value of 328 lux⋅seconds) compensates for reflections off light colored ceilings, which is typical for indoor flash photography.
Outdoors mixing of continuous-light and GN-based fill flash
This all means that in outdoor settings, flash devices must be roughly nine-percent closer ${\textstyle ({\sqrt {270/328}}=90.7\%)}$ than their indoor-rated guide numbers indicate in order to obtain a full-rated luminous exposure. The above fill-flash tables give distances that have been adjusted accordingly.
Also bear in mind that the distances given in the above tables do not obey the f‑number × distance formula when using the f‑stops shown in the first column because the fill-flash contribution is one f‑stop less than the sunlit portions of the scene—a 50% underexposure. The values may be correctly calculated by hand when the apertures shown in the first column are larger (numerically smaller f‑number) by one f‑stop. Those larger apertures, when expressed in industry-standard nomenclature, are f/11, f/8, f/6.3 (f/5.6 +⅓), and f/5.6 but have the following more precise values underlying the above tables: f/11.3137, f/8, f/6.3496, and f/5.6569 ${\textstyle {({\sqrt {2}}}^{\ 7}}$ , ${\textstyle {\sqrt {2}}^{\ 6}}$ , ${\textstyle {\sqrt {2}}^{\ 5{\tfrac {1}{3}}}}$ , and ${\textstyle {\sqrt {2}}^{\ 5})}$ .

metz-mecatech.de

One notable way of expressing guide numbers scaled for use with both feet and meters (and relative to an ISO setting of 100) is as practiced by Metz mecatech GmbH, as exemplified by the data sheet for their mecablitz 52 AF-1 digital (product page here):
Flash output
• High max. guide number 52 (meters), High max. guide
number 170 (feet) for ISO 100/21° and 105 mm
Another notable example is Nikon's practice for expressing the technical specifications of products marketed in the U.S., such as their SB-5000 AF Speedlight (product page here):
Guide Number
34.5 m/113 ft. (at 35 mm) 55
m/180 ft. (at 200 mm) (FX
format, standard illumination
pattern) (at ISO 100)
Yet another notable example is Canon USA's practices, as exemplified by their Speedlite 430EX III-RT (product page here), which is as follows:
Guide Number The maximum Guide No. is approximately 141.1 ft./43m at ISO 100 and 105 mm flash coverage.
The style for expressing guide numbers throughout the metric-observing world is typified by the practices of Metz mecatecheh GmbH, as exemplified by the German-language version of their website for their mecablitz 76 MZ-5 digital (product page specifications here):
Blitzleistung
• Hohe max. Leitzahl 76 bei ISO 100/21° und 105 mm

This translates to English as follows:

Flash output
• High max. guide number 76 at ISO 100/21° and 105 mm

nikonusa.com

One notable way of expressing guide numbers scaled for use with both feet and meters (and relative to an ISO setting of 100) is as practiced by Metz mecatech GmbH, as exemplified by the data sheet for their mecablitz 52 AF-1 digital (product page here):
Flash output
• High max. guide number 52 (meters), High max. guide
number 170 (feet) for ISO 100/21° and 105 mm
Another notable example is Nikon's practice for expressing the technical specifications of products marketed in the U.S., such as their SB-5000 AF Speedlight (product page here):
Guide Number
34.5 m/113 ft. (at 35 mm) 55
m/180 ft. (at 200 mm) (FX
format, standard illumination
pattern) (at ISO 100)
Yet another notable example is Canon USA's practices, as exemplified by their Speedlite 430EX III-RT (product page here), which is as follows:
Guide Number The maximum Guide No. is approximately 141.1 ft./43m at ISO 100 and 105 mm flash coverage.
Nikon, for instance, gives two ratings when providing the technical specifications for their SB-910 AF Speedlight, one of which is relative to ISO 200 (product page here):
Guide Number
34 m/111.5 ft. (at ISO 100,
35mm zoom head position, in
FX format, standard
illumination pattern,
20°C/68°F) to 48 m/157.5 ft.
(at ISO 200, 35 mm zoom
head position, in FX format,
standard illumination pattern,
20°C/68°F)

scantips.com

Scantips.com: Understanding Flash Guide Numbers, plus GN Calculator

sekonic.com

The continuous-light portion of fill-flash photos
The continuous-light exposures underlying the above tables are in accordance with Eastman Kodak Company's guidelines for exposing film. For example, as exemplified by the technical data sheet for their Kodak Professional Ektar 100 film (PDF here), scenes that are lit by bright or hazy sun with distinct shadows and which are shot between 2 hours after sunrise and 2 hours before sunset are an illuminance of 40,960 lux Archived 2017-08-14 at the Wayback Machine (EV 14 @ ISO 100). For this illuminance, Kodak recommends settings (ISO 100, f/11, ¹/₁₂₅ th of a second) that equal a luminous exposure of 328 lux⋅seconds. This exposure level applies proportionally across the full gamut of Kodak's color and panchromatic B&W films where—at 40,960 lux and f/11—a film speed of ISO 200 is properly exposed when a scene receives a luminous exposure of 164 lux⋅seconds, ISO 400 requires 82 lux⋅seconds, and so forth. Kodak's guidance for a proper continuous-light exposure for all film speeds, scene illuminance, and camera settings reduces as follows:

${\frac {E_{v}\cdot t\cdot S}{f^{2}}}=k$

…where,

E_v is scene illuminance in lux (e.g. 40,960; 81,900; and 111,000). A table of lux values vs. EV @ ISO 100 is available here, by Sekonic. Archived 2017-08-14 at the Wayback Machine Archived 2017-08-14 at the Wayback Machine

t is exposure time in seconds (e.g. 0.008 for ¹/₁₂₅ th of a second)

S is ISO arithmetic speed or sensitivity (e.g. 100, 200, and 400)

f is the f-number (e.g. 8 for an f‑stop of f/8. Common f‑stops from the ${\textstyle {\sqrt {2}}^{\ n}}$ series include f/1.1892, f/1.4142, f/2, f/2.8284, f/4, f/5.6569, f/8, f/11.3137, f/16, f/22.6274, and f/32)

k is a constant equaling 256

When utilizing fill flash, where balancing flash and continuous light can be difficult, the following four derivatives of this continuous-light exposure equation can be useful:

${\sqrt {\frac {E_{v}\cdot t\cdot S}{k}}}=f$

${\frac {{f^{2}}\cdot {k}}{E_{v}\cdot t}}=S$

${\frac {{f^{2}}\cdot {k}}{t\cdot S}}=E_{v}$

${\frac {{f^{2}}\cdot {k}}{E_{v}\cdot S}}=t$ (take ⁠1/1⁠ to obtain the denominator of the fractional shutter speed)

For any combination of lighting, film, and camera settings that conforms to one of the above five equations, a proper luminous exposure is calculated as follows:

$E_{v}\cdot t=H_{v}$

…where,

E_v is scene illuminance in lux (e.g. 40,960; 81,900; and 111,000)

t is exposure time in seconds (e.g. 0.008 for ¹/₁₂₅ th of a second)

H_v is luminous exposure in lux⋅seconds

Note that Kodak's exposure guidelines—for photographs taken in typical settings without the benefit of incident-light meters—are for pictures shot during a broad portion of the day with even some light haze in the sky; this is half as bright as the clear-sky, near-noon, open-area, "sunny f/16 rule", which is EV 15 at ISO 100, or 81,900 lux. Notwithstanding differences in the assumed daylight illuminance, a proper continuous-light exposure is still governed by the same mathematical relationship where (E_v⋅t⋅S)/f² = k.
Flash exposures vs. continuous-light exposures
Importantly, Eastman Kodak Company's exposure recommendations for electronic flash are slightly different from those for continuous-light exposures. Kodak calculates exposures for electronic flash devices according to the following formula:

$GN={\sqrt {\frac {BCPS\times ISO}{20}}}$

…where,

GN is guide number

BCPS is beam candlepower seconds

ISO is ISO arithmetic speed or sensitivity (e.g. 100, 200, and 400)

See Characterizing the Output of Photographic Flash Units, by Douglas A. Kerr (PDF here) for more.

This formula means that under the same conditions used above for Kodak Professional Ektar 100 film (ISO 100 and f/11), a proper electronic flash exposure equals only 270 lux⋅seconds (one candlepower equals 0.981 candela, which equals 0.981 lux⋅second at a distance of one meter). This slightly reduced flash exposure (versus the continuous-light value of 328 lux⋅seconds) compensates for reflections off light colored ceilings, which is typical for indoor flash photography.
Outdoors mixing of continuous-light and GN-based fill flash
This all means that in outdoor settings, flash devices must be roughly nine-percent closer ${\textstyle ({\sqrt {270/328}}=90.7\%)}$ than their indoor-rated guide numbers indicate in order to obtain a full-rated luminous exposure. The above fill-flash tables give distances that have been adjusted accordingly.
Also bear in mind that the distances given in the above tables do not obey the f‑number × distance formula when using the f‑stops shown in the first column because the fill-flash contribution is one f‑stop less than the sunlit portions of the scene—a 50% underexposure. The values may be correctly calculated by hand when the apertures shown in the first column are larger (numerically smaller f‑number) by one f‑stop. Those larger apertures, when expressed in industry-standard nomenclature, are f/11, f/8, f/6.3 (f/5.6 +⅓), and f/5.6 but have the following more precise values underlying the above tables: f/11.3137, f/8, f/6.3496, and f/5.6569 ${\textstyle {({\sqrt {2}}}^{\ 7}}$ , ${\textstyle {\sqrt {2}}^{\ 6}}$ , ${\textstyle {\sqrt {2}}^{\ 5{\tfrac {1}{3}}}}$ , and ${\textstyle {\sqrt {2}}^{\ 5})}$ .

web.archive.org

The continuous-light portion of fill-flash photos
The continuous-light exposures underlying the above tables are in accordance with Eastman Kodak Company's guidelines for exposing film. For example, as exemplified by the technical data sheet for their Kodak Professional Ektar 100 film (PDF here), scenes that are lit by bright or hazy sun with distinct shadows and which are shot between 2 hours after sunrise and 2 hours before sunset are an illuminance of 40,960 lux Archived 2017-08-14 at the Wayback Machine (EV 14 @ ISO 100). For this illuminance, Kodak recommends settings (ISO 100, f/11, ¹/₁₂₅ th of a second) that equal a luminous exposure of 328 lux⋅seconds. This exposure level applies proportionally across the full gamut of Kodak's color and panchromatic B&W films where—at 40,960 lux and f/11—a film speed of ISO 200 is properly exposed when a scene receives a luminous exposure of 164 lux⋅seconds, ISO 400 requires 82 lux⋅seconds, and so forth. Kodak's guidance for a proper continuous-light exposure for all film speeds, scene illuminance, and camera settings reduces as follows:

${\frac {E_{v}\cdot t\cdot S}{f^{2}}}=k$

…where,

E_v is scene illuminance in lux (e.g. 40,960; 81,900; and 111,000). A table of lux values vs. EV @ ISO 100 is available here, by Sekonic. Archived 2017-08-14 at the Wayback Machine Archived 2017-08-14 at the Wayback Machine

t is exposure time in seconds (e.g. 0.008 for ¹/₁₂₅ th of a second)

S is ISO arithmetic speed or sensitivity (e.g. 100, 200, and 400)

f is the f-number (e.g. 8 for an f‑stop of f/8. Common f‑stops from the ${\textstyle {\sqrt {2}}^{\ n}}$ series include f/1.1892, f/1.4142, f/2, f/2.8284, f/4, f/5.6569, f/8, f/11.3137, f/16, f/22.6274, and f/32)

k is a constant equaling 256

When utilizing fill flash, where balancing flash and continuous light can be difficult, the following four derivatives of this continuous-light exposure equation can be useful:

${\sqrt {\frac {E_{v}\cdot t\cdot S}{k}}}=f$

${\frac {{f^{2}}\cdot {k}}{E_{v}\cdot t}}=S$

${\frac {{f^{2}}\cdot {k}}{t\cdot S}}=E_{v}$

${\frac {{f^{2}}\cdot {k}}{E_{v}\cdot S}}=t$ (take ⁠1/1⁠ to obtain the denominator of the fractional shutter speed)

For any combination of lighting, film, and camera settings that conforms to one of the above five equations, a proper luminous exposure is calculated as follows:

$E_{v}\cdot t=H_{v}$

…where,

E_v is scene illuminance in lux (e.g. 40,960; 81,900; and 111,000)

t is exposure time in seconds (e.g. 0.008 for ¹/₁₂₅ th of a second)

H_v is luminous exposure in lux⋅seconds

Note that Kodak's exposure guidelines—for photographs taken in typical settings without the benefit of incident-light meters—are for pictures shot during a broad portion of the day with even some light haze in the sky; this is half as bright as the clear-sky, near-noon, open-area, "sunny f/16 rule", which is EV 15 at ISO 100, or 81,900 lux. Notwithstanding differences in the assumed daylight illuminance, a proper continuous-light exposure is still governed by the same mathematical relationship where (E_v⋅t⋅S)/f² = k.
Flash exposures vs. continuous-light exposures
Importantly, Eastman Kodak Company's exposure recommendations for electronic flash are slightly different from those for continuous-light exposures. Kodak calculates exposures for electronic flash devices according to the following formula:

$GN={\sqrt {\frac {BCPS\times ISO}{20}}}$

…where,

GN is guide number

BCPS is beam candlepower seconds

ISO is ISO arithmetic speed or sensitivity (e.g. 100, 200, and 400)

See Characterizing the Output of Photographic Flash Units, by Douglas A. Kerr (PDF here) for more.

This formula means that under the same conditions used above for Kodak Professional Ektar 100 film (ISO 100 and f/11), a proper electronic flash exposure equals only 270 lux⋅seconds (one candlepower equals 0.981 candela, which equals 0.981 lux⋅second at a distance of one meter). This slightly reduced flash exposure (versus the continuous-light value of 328 lux⋅seconds) compensates for reflections off light colored ceilings, which is typical for indoor flash photography.
Outdoors mixing of continuous-light and GN-based fill flash
This all means that in outdoor settings, flash devices must be roughly nine-percent closer ${\textstyle ({\sqrt {270/328}}=90.7\%)}$ than their indoor-rated guide numbers indicate in order to obtain a full-rated luminous exposure. The above fill-flash tables give distances that have been adjusted accordingly.
Also bear in mind that the distances given in the above tables do not obey the f‑number × distance formula when using the f‑stops shown in the first column because the fill-flash contribution is one f‑stop less than the sunlit portions of the scene—a 50% underexposure. The values may be correctly calculated by hand when the apertures shown in the first column are larger (numerically smaller f‑number) by one f‑stop. Those larger apertures, when expressed in industry-standard nomenclature, are f/11, f/8, f/6.3 (f/5.6 +⅓), and f/5.6 but have the following more precise values underlying the above tables: f/11.3137, f/8, f/6.3496, and f/5.6569 ${\textstyle {({\sqrt {2}}}^{\ 7}}$ , ${\textstyle {\sqrt {2}}^{\ 6}}$ , ${\textstyle {\sqrt {2}}^{\ 5{\tfrac {1}{3}}}}$ , and ${\textstyle {\sqrt {2}}^{\ 5})}$ .