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Re: [Phys-L] triangular induction puzzle



okay, I've got '14' (and a second way of getting '12' ) but really should be getting on now…

On 12/10/2022 11:59, Prof. Keith S. Taber wrote:
I can also offer '11'…

K


On 12/10/2022 11:51, Prof. Keith S. Taber wrote:
I should clarify

I have '12' as an alternative answer.

I do NOT have twelve alternative answers.

K

On 12/10/2022 11:48, Prof. Keith S. Taber wrote:
That is certainly a justifiable answer (and I admit the only one I readily spotted) but how can we know that was the answer the puzzle solver had in mind?

I am sure we can find alternatives. I have 12, for example.

Keith


--

“there is no ‘scientific worldview’ just as there is no uniform enterprise ‘science’ - except in the minds of metaphysicians, schoolmasters, and scientists blinded by the achievements of their own particular niche” <https://eur03.safelinks.protection.outlook.com/?url=https%3A%2F%2Fscience-education-research.com%2Fcommonplace%2F&data=05%7C01%7Ckst24%40cam.ac.uk%7Ceb17bcd42a0447036a8608daac40e1c4%7C49a50445bdfa4b79ade3547b4f3986e9%7C0%7C0%7C638011691904330975%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=94teZ%2FCoSC4%2FokPNsI60yqBmA62XUmdoJRt9vPaCVgk%3D&reserved=0>


Dr. Keith S. Taber

Emeritus Professor of Science Education
University of Cambridge
http://www.educ.cam.ac.uk/staff/taber.html


Senior Member
Homerton College, Cambridge


https://science-education-research.com



Some recent postings include

Falsifying research conclusions. You do not need to falsify your results if you are prepared to draw conclusions contrary to the outcome of your data analysis. <https://eur03.safelinks.protection.outlook.com/?url=https%3A%2F%2Fscience-education-research.com%2Ffalsifying-research-conclusions%2F&data=05%7C01%7Ckst24%40cam.ac.uk%7Ceb17bcd42a0447036a8608daac40e1c4%7C49a50445bdfa4b79ade3547b4f3986e9%7C0%7C0%7C638011691904330975%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=hug4xvSxlElteKDY3E922nDXFrdzWoBE%2F6MmPEOlP%2F8%3D&reserved=0>

Delusions of educational impact. A 'peer-reviewed' study claims to improve academic performance by purifying the souls of students suffering from hallucinations <https://eur03.safelinks.protection.outlook.com/?url=https%3A%2F%2Fscience-education-research.com%2Fdelusions-of-educational-impact%2F&data=05%7C01%7Ckst24%40cam.ac.uk%7Ceb17bcd42a0447036a8608daac40e1c4%7C49a50445bdfa4b79ade3547b4f3986e9%7C0%7C0%7C638011691904330975%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=53SpyHr%2FB3EHu%2B2P1prsbnqECjF3d9OnA%2BdpoYTeCEE%3D&reserved=0>

Can deforestation stop indigenous groups starving? <https://eur03.safelinks.protection.outlook.com/?url=https%3A%2F%2Fscience-education-research.com%2Fcan-deforestation-stop-indigenous-groups-starving%2F&data=05%7C01%7Ckst24%40cam.ac.uk%7Ceb17bcd42a0447036a8608daac40e1c4%7C49a50445bdfa4b79ade3547b4f3986e9%7C0%7C0%7C638011691904330975%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=cHuPOvYx0JPr75yWhxpf1xRF%2FOh%2FeaByuzRetzFf%2Bjs%3D&reserved=0>

How much damage can a couple of molecules do? Just how dangerous is Novichok? <https://eur03.safelinks.protection.outlook.com/?url=https%3A%2F%2Fscience-education-research.com%2Fhow-much-damage-can-a-couple-of-molecules-do%2F&data=05%7C01%7Ckst24%40cam.ac.uk%7Ceb17bcd42a0447036a8608daac40e1c4%7C49a50445bdfa4b79ade3547b4f3986e9%7C0%7C0%7C638011691904330975%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=vrhQT9OYSUwsLBM0IQ5W0JtK8V6PVJ2wiBRC%2B%2BxXeW4%3D&reserved=0>

Of mostly natural origin. Is your shampoo of natural, unnatural, or supernatural origin? <“https://science-education-research.com/of-mostly-natural-origin/”;>

Bats are [almost certainly] not closely related to viruses. <“https://science-education-research.com/bats-are-almost-certainly-not-closely-related-to-viruses/”;>





Recently published scholarship:

*Taber, K. S. (2019). Experimental research into teaching innovations: responding to methodological and ethical challenges. *
/Studies in Science Education./ doi:10.1080/03057267.2019.1658058

*/The Nature of the Chemical Concept/*/: Re-constructing Chemical Knowledge in Teaching and Learning/
Royal Society of Chemistry, https://pubs.rsc.org/en/content/ebook/978-1-78262-460-8

*Taber, K. S. (2021). The Challenge to Educational Reforms during a Global Emergency: The Case of Progressive Science Education./C.E.P.S. Journal/, 11 (Special issue), 1-21. doi:doi: 10.26529/cepsj.1109*


Recent publications primarily for teachers:

*/Foundations for Teaching Chemistry/*/: Chemical knowledge for teaching/
Routledge, ISBN: 9780815377740

*/MasterClass in Science Education/*/: Transforming Teaching and Learning/
Bloomsbury Academic, ISBN: 9781474289412


Editor-in-Chief: */RSC Advances in Chemistry Education Series/*
Royal Society of Chemistry Book Series
Proposals for Advances in Chemistry Education volumes (edited or monograph) are invited. Scholarly books on all aspects of chemistry education are being considered for publication in the series ( https://science-education-research.com/advances-in-chemistry-education/ )




*ECLIPSE - /Exploring Conceptual Learning, Integration and Progression in Science Education/*
https://science-education-research.com/projects/eclipse/



*Reject publisher contracts that ask you to waive your legal rights*
Authors are granted the 'moral' right to protect the integrity of their works in law.
There is no reason for reputable academic publishers to ask authors to allow them to be able to make changes of any kind, at any time, to authors' works, without the author's approval.
Scholars should not be prepared to abandon their legal right to object to changes that undermine the integrity of their scholarly works.
Academics should refuse to sign publishers' contracts that include clauses that require authors to waive their legal moral rights.
Sign a petition to ask publishers to respect authors' legal rights: *https://science-education-research.com/academic-standards/defend-the-moral-right-to-the-integrity-of-your-scholarly-work/*



--

“there is no ‘scientific worldview’ just as there is no uniform enterprise ‘science’ - except in the minds of metaphysicians, schoolmasters, and scientists blinded by the achievements of their own particular niche” <https://eur03.safelinks.protection.outlook.com/?url=https%3A%2F%2Fscience-education-research.com%2Fcommonplace%2F&data=05%7C01%7Ckst24%40cam.ac.uk%7Ceb17bcd42a0447036a8608daac40e1c4%7C49a50445bdfa4b79ade3547b4f3986e9%7C0%7C0%7C638011691904486941%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=MuGPgVsVtBYuzEjz%2Bt5XnNalcP8BzWhocp4js5Uhw5I%3D&reserved=0>


Dr. Keith S. Taber

Emeritus Professor of Science Education
University of Cambridge
http://www.educ.cam.ac.uk/staff/taber.html


Senior Member
Homerton College, Cambridge


https://science-education-research.com



Some recent postings include

Falsifying research conclusions. You do not need to falsify your results if you are prepared to draw conclusions contrary to the outcome of your data analysis. <https://eur03.safelinks.protection.outlook.com/?url=https%3A%2F%2Fscience-education-research.com%2Ffalsifying-research-conclusions%2F&data=05%7C01%7Ckst24%40cam.ac.uk%7Ceb17bcd42a0447036a8608daac40e1c4%7C49a50445bdfa4b79ade3547b4f3986e9%7C0%7C0%7C638011691904486941%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=%2Bc4Q7FKngmqMHYVjwTvAi4R0uLj4rQTia2NkSWlp5HQ%3D&reserved=0>

Delusions of educational impact. A 'peer-reviewed' study claims to improve academic performance by purifying the souls of students suffering from hallucinations <https://eur03.safelinks.protection.outlook.com/?url=https%3A%2F%2Fscience-education-research.com%2Fdelusions-of-educational-impact%2F&data=05%7C01%7Ckst24%40cam.ac.uk%7Ceb17bcd42a0447036a8608daac40e1c4%7C49a50445bdfa4b79ade3547b4f3986e9%7C0%7C0%7C638011691904486941%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=nU3newam7mqEHymZurx2El5%2BLtoyQMFjgaMIsbOu6vI%3D&reserved=0>

Can deforestation stop indigenous groups starving? <https://eur03.safelinks.protection.outlook.com/?url=https%3A%2F%2Fscience-education-research.com%2Fcan-deforestation-stop-indigenous-groups-starving%2F&data=05%7C01%7Ckst24%40cam.ac.uk%7Ceb17bcd42a0447036a8608daac40e1c4%7C49a50445bdfa4b79ade3547b4f3986e9%7C0%7C0%7C638011691904486941%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=zK9hjkNPYSLg3O6vVNFp0nJ3WYOFDGFqki2FF385UDQ%3D&reserved=0>

How much damage can a couple of molecules do? Just how dangerous is Novichok? <https://eur03.safelinks.protection.outlook.com/?url=https%3A%2F%2Fscience-education-research.com%2Fhow-much-damage-can-a-couple-of-molecules-do%2F&data=05%7C01%7Ckst24%40cam.ac.uk%7Ceb17bcd42a0447036a8608daac40e1c4%7C49a50445bdfa4b79ade3547b4f3986e9%7C0%7C0%7C638011691904486941%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=S3MPOAx%2F2mguXxRsYvYL7kj0fcwEW4j9CY8qrsVhaHU%3D&reserved=0>

Of mostly natural origin. Is your shampoo of natural, unnatural, or supernatural origin? <“https://science-education-research.com/of-mostly-natural-origin/”;>

Bats are [almost certainly] not closely related to viruses. <“https://science-education-research.com/bats-are-almost-certainly-not-closely-related-to-viruses/”;>





Recently published scholarship:

*Taber, K. S. (2019). Experimental research into teaching innovations: responding to methodological and ethical challenges. *
/Studies in Science Education./ doi:10.1080/03057267.2019.1658058

*/The Nature of the Chemical Concept/*/: Re-constructing Chemical Knowledge in Teaching and Learning/
Royal Society of Chemistry, https://pubs.rsc.org/en/content/ebook/978-1-78262-460-8

*Taber, K. S. (2021). The Challenge to Educational Reforms during a Global Emergency: The Case of Progressive Science Education./C.E.P.S. Journal/, 11 (Special issue), 1-21. doi:doi: 10.26529/cepsj.1109*


Recent publications primarily for teachers:

*/Foundations for Teaching Chemistry/*/: Chemical knowledge for teaching/
Routledge, ISBN: 9780815377740

*/MasterClass in Science Education/*/: Transforming Teaching and Learning/
Bloomsbury Academic, ISBN: 9781474289412


Editor-in-Chief: */RSC Advances in Chemistry Education Series/*
Royal Society of Chemistry Book Series
Proposals for Advances in Chemistry Education volumes (edited or monograph) are invited. Scholarly books on all aspects of chemistry education are being considered for publication in the series ( https://science-education-research.com/advances-in-chemistry-education/ )




*ECLIPSE - /Exploring Conceptual Learning, Integration and Progression in Science Education/*
https://science-education-research.com/projects/eclipse/



*Reject publisher contracts that ask you to waive your legal rights*
Authors are granted the 'moral' right to protect the integrity of their works in law.
There is no reason for reputable academic publishers to ask authors to allow them to be able to make changes of any kind, at any time, to authors' works, without the author's approval.
Scholars should not be prepared to abandon their legal right to object to changes that undermine the integrity of their scholarly works.
Academics should refuse to sign publishers' contracts that include clauses that require authors to waive their legal moral rights.
Sign a petition to ask publishers to respect authors' legal rights: *https://science-education-research.com/academic-standards/defend-the-moral-right-to-the-integrity-of-your-scholarly-work/*



--

“there is no ‘scientific worldview’ just as there is no uniform enterprise ‘science’ - except in the minds of metaphysicians, schoolmasters, and scientists blinded by the achievements of their own particular niche” <https://science-education-research.com/commonplace/>


Dr. Keith S. Taber

Emeritus Professor of Science Education
University of Cambridge
http://www.educ.cam.ac.uk/staff/taber.html


Senior Member
Homerton College, Cambridge


https://science-education-research.com



Some recent postings include

Falsifying research conclusions. You do not need to falsify your results if you are prepared to draw conclusions contrary to the outcome of your data analysis. <https://science-education-research.com/falsifying-research-conclusions/>

Delusions of educational impact. A 'peer-reviewed' study claims to improve academic performance by purifying the souls of students suffering from hallucinations <https://science-education-research.com/delusions-of-educational-impact/>

Can deforestation stop indigenous groups starving? <https://science-education-research.com/can-deforestation-stop-indigenous-groups-starving/>

How much damage can a couple of molecules do? Just how dangerous is Novichok? <https://science-education-research.com/how-much-damage-can-a-couple-of-molecules-do/>

Of mostly natural origin. Is your shampoo of natural, unnatural, or supernatural origin? <“https://science-education-research.com/of-mostly-natural-origin/”;>

Bats are [almost certainly] not closely related to viruses. <“https://science-education-research.com/bats-are-almost-certainly-not-closely-related-to-viruses/”;>





Recently published scholarship:

*Taber, K. S. (2019). Experimental research into teaching innovations: responding to methodological and ethical challenges. *
/Studies in Science Education./ doi:10.1080/03057267.2019.1658058

*/The Nature of the Chemical Concept/*/: Re-constructing Chemical Knowledge in Teaching and Learning/
Royal Society of Chemistry, https://pubs.rsc.org/en/content/ebook/978-1-78262-460-8

*Taber, K. S. (2021). The Challenge to Educational Reforms during a Global Emergency: The Case of Progressive Science Education./C.E.P.S. Journal/, 11 (Special issue), 1-21. doi:doi: 10.26529/cepsj.1109*


Recent publications primarily for teachers:

*/Foundations for Teaching Chemistry/*/: Chemical knowledge for teaching/
Routledge, ISBN: 9780815377740

*/MasterClass in Science Education/*/: Transforming Teaching and Learning/
Bloomsbury Academic, ISBN: 9781474289412


Editor-in-Chief: */RSC Advances in Chemistry Education Series/*
Royal Society of Chemistry Book Series
Proposals for Advances in Chemistry Education volumes (edited or monograph) are invited. Scholarly books on all aspects of chemistry education are being considered for publication in the series ( https://science-education-research.com/advances-in-chemistry-education/ )




*ECLIPSE - /Exploring Conceptual Learning, Integration and Progression in Science Education/*
https://science-education-research.com/projects/eclipse/



*Reject publisher contracts that ask you to waive your legal rights*
Authors are granted the 'moral' right to protect the integrity of their works in law.
There is no reason for reputable academic publishers to ask authors to allow them to be able to make changes of any kind, at any time, to authors' works, without the author's approval.
Scholars should not be prepared to abandon their legal right to object to changes that undermine the integrity of their scholarly works.
Academics should refuse to sign publishers' contracts that include clauses that require authors to waive their legal moral rights.
Sign a petition to ask publishers to respect authors' legal rights: *https://science-education-research.com/academic-standards/defend-the-moral-right-to-the-integrity-of-your-scholarly-work/*