Here's how to solve the 'simple' high-school math problem stumping everyone on the internet

A question from a global math Olympiad for high-school children in Singapore is going viral after the question was leaked online.

The question seems simple — all you need to do is work out Cheryl's birthday, based on three statements and with just 10 possible dates to chose from — but the logic to solving it will really make your brain crinkle.

First, here's the question so you can have a go for yourself:

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Here's the question in plain text:

Albert and Bernard just became friends with Cheryl, and they want to know when her birthday is. Cheryl gives them a list of 10 possible dates: May 15, May 16, May 19, June 17, June 18, July 14, July 16, August 14, August 15, August 17.

Cheryl then tells Albert and Bernard separately the month and the day of her birthday, respectively.

Albert: I don’t know when Cheryl’s birthday is, but I know that Bernard does not know too.

Bernard: At first I don’t know when Cheryl’s birthday is, but I know now.

Albert: Then I also know when Cheryl’s birthday is.

So when is Cheryl’s birthday?

Think you know the answer? The question prompted so much confusion that the creators of the test posted an explanation on Facebook to clarify matters. 

The official solution is below. However, some commenters have found the logic even more confusing than the question itself, so we've broken it down even further below. 

Singapore and Asian School Math Olympiads/Facebook

If that solution confused you, we've broken it down piece by piece here:

• At the start, Albert and Bernard are each given one piece of information about Cheryl's birthday. Albert is given the month and Bernard is given the day. You have to assume that Cheryl was smart enough not give either of them a day or month that would enable them to work out the full date on their own.
• Now back to Bernard. Bernard knows the day (but not the month) the birthday falls on. We can rule out May 19 and June 18 straightaway because these days (18 and 19) occur only once in the list of dates. In contrast, 14, 15, 16, and 17 all appear twice. So, if Bernard was told either 18 or 19, he would be able to infer the month by knowing only the day. And that would be a different puzzle. So we can cross those two dates off the list.

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• But Albert knows that Bernard does not initially know the birthday. How? Well, Albert knows the month it falls in. Had Cheryl told Albert May or June (the only two months with unique number dates) then it's possible (but not guaranteed) that Bernard already knows the full date. But Albert knows that Bernard doesn't know the full date, which means Cheryl must have told Albert that the month is either July or August. So all other months get crossed off the list.

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• Bernard realises what the date is after Albert first says that he knowsthat Bernard doesn't know it. How does this work? Once Albert says that Bernard doesn't know, it shows to Bernard that it can't be in May or June. (After all, if it was, then Albert wouldn't know that Bernard doesn't know.) So this leaves July and August. Now, remember that Bernard knows the day (but not the month). Of the 5 options, 14 is the only one repeated twice. If it were July 14 or August 14 he still wouldn't know — but he does know, which means the day he was told must have been 15, 16, or 17, and we can cross another two possibilities out.

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• And then, after Bernard realises what the date is, Albert follows suit. This is because — as we've established — Albert knows only the month. With 3 options left, if Albert had been told it were in August, he still wouldn't know, because there are 2 August options left. But that's not the case — Albertdoes know with certainty what date it is. From this we can infer that it isJuly 16, as there's only one July option there.

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To recap, Albert was told that Cheryl's birthday was in July, while Bernard was told it landed on the 17. This limited knowledge allowed each to narrow down their options — and by making inferences from the other's ignorance, the two could continue to thin down the possible categories until there was only one option left.

By now, many people have chimed in to offer their own way of solving this math elimination problem and even an alternative answer of August 17 — to which the Singapore and Asian School Math Olympiads has also responded.

"If there are two possible answers, then this problem is ambiguous," the organisation said in a Facebook post.

Still, there are others who are willing to find the humour in all of this. One commenter writes: "I personally know Cheryl and her birthday is May 19."

7-S Framework McKinsey

Definition: 7-S Framework McKinsey

The McKinsey 7- S framework is a popular model used in organizations to analyse the environment to investigate if the company is achieving its intended objectives.

The name of the model can be explained by the fact that the model was developed by Tom Peters & Robert Waterman, consultants at the McKinsey & Company consulting firm.

The model proposes 7 interdependent factors – 3 hard ‘S’ i.e. strategy, structure, systems; and 4 soft ‘S’ i.e. shared values, skills, style and staff.

The hard ‘S’ are more tangible, easily to define and easy to influence than the soft ‘S’.

Strategy – It refers to the intended sequence of actions taken by a company to achieve its goals and objectives. It deals with resource allocation and includes competition, customers and the environment.
Structure – It refers to how the various business units are structured and how they communicate with each other. A company’s structure may be centralized or decentralized or may take many other forms depending on the company’s culture and values.

Systems – This includes a host of systems within an organization that define its processes and routines. It includes performance appraisal system, financial systems, IT systems etc.

Shared values – These are the core values of the company that connect all the other 6 factors. These are the fundamental ideas or guiding principles that lay the foundation of businesses.

Skills – These define the core competencies of the employees.

Style – This spans the core beliefs, norms and management style in the organization.

Staff – It refers to the number and type of employees in the organization. It is very important for an organization to manage its human capital to create competitive advantage.

Marketing: Brand Portfolio: Defination

Definition: Brand Portfolio

The brand portfolio of a company is the complete range of all brands and brand lines it offers for sale in a particular category or market segment. A brand portfolio is said to be optimal when each brand in it has the ability to maximize equity in combination with all the other brands.

The basic principle of a brand portfolio is to maximize market coverage and to offer enough brands so that no potential customers feel left out, but at the same time to minimize overlap so that no two brands cannibalize each other. A good brand portfolio is marked by the clear differentiation of the component brands ad economy (Marketing and production costs are justified by the size of the end customer segment).

Brand portfolios must be continuously pruned, so as to identify weak brands, reenergize them or to kill off the unprofitable ones.

Within a portfolio, brands can play the following roles:

• Flankers (or fighters): Positioned with respect to the competitors’ brands so that the flagship or more important brands are protected. Care should be taken so as not to cannibalize the flagship brand.

• Cash cows: these may be showing falling sales but still command decent profits despite less marketing support.

• Low-end entry level: Relatively low-priced brands designed to attract the first-time consumer to the brand franchise. These customers can later be “traded up’ to higher-priced brands.

• High-end prestige: Higher-priced brands to add prestige and credibility to the entire portfolio

TOP MBA JOB INTERVIEW QUESTIONS AND ANSWERS

Interviewing a candidate for a job is a tough task not only for the a candidate but also for the recruiter. Each one of them is at their best to prove their worth as a potential employee. Yet some cross the line and get selected to get the coveted job. Here are the basic questions you’re most likely to be asked in a job interview:

1) Question: Tell me about yourself and your background

Answer: The first question that a candidate is suppose to answer. The best way to tackle such a question is so quickly run the panel through your education, your achievements in academics, previous work experience (if you have worked before), exceptional feats, extra-curricular achievements etc. Such an explanation is a chronological manner gives a systematic approach to the interview

2) Question: Why have you applied for this position and job opening?

Answer: Recruiters need to see that you have a complete idea about the job decription and the skill sets required for the position you are applying for. Hence, they want to understand your knowledge about the profile and whether this job was your preference or not

3) Question: What knowledge do you have about our company?

Answer: Until and unless you are thorough about the company you are applying for, chances of selection are quite bleak. Remember, organisations are looking for skilled resources to build their company, each revenue and also select potential leaders who stay long in the organisation

4) Question: Why did you quit your last job? or Why are you still a fresher?

Answer: The reason behind you change of job is pivotal. The company wants to know whether you were an asset to you previous recruiter and that you are changing at your will for better avenues. However, if you have been forced to leave your previous job due to personal reasons or failing to meet company expectations, its gives a bad impression. Also, freshers are grilled as to why they haven't taken "efforts" to enhance their professional skill sets by taking up some job before completing higher studies.

5) Question: What job duties, responsibilities, initiatives etc have you taken?

Answer: This is a critical answer as far as the recruiter understanding your cohesion with the job profile being offered. Be short, crisp and confident in answering such a question. The best way is to break up the answer into three: (a) problem statement which you worked on; (b) your understanding, analysis, actions, execution (c) end result and how it benefited the company revenue or operations

6) Question: Tell me about your Strengths and Weaknesses

Answer: Most candidates don't understand the seriousness of this question but then this carries a lot of weightage as recruiters know that how well do candidates know themselves. Whatever strengths are described need to be backed up by some experience, else it appears to be only worthless statements. Weaknesses is a question which most people fail to tackle and end up saying 'I don't have weaknesses'. Every human being has some weak points whether they are lazy, lack certain skills, less attentive, over-emotional etc. Whatever the weaknesses are, mention that you are working to rectify them consciously.

7) Question: Give an example where you handled a team, managed a crisis etc

Answer: Such a question tells about the responsibilities and maturity levels that a candidate is able to handle. All the above questions are a part of an everyday life of a budding manager. Hence, real-life experiences of the candidate add more spine to the education and work-experience.

8) Question: Where do you see yourself 5 years from now?

Answer: This answer tells how ambitious and career-oriented the candidate is. A correct answer can show the career path the candidate wants to achieve by gradually growing at the companys' hierarchial pace. If you want to become an entrepreneur, then mention that but reiterate that for that you would want to gain atleast 3 to 5 years of professional work experience. This confidence shows that you are mentally prepared and respect you career choices as well.

9) Question: What salary range are you looking for?

Answer: The salary structure is the most important discussion that happens in the interview. A salary discussion is a positive sign but yet it is not affirmative. A candidate must quote a confident estimate that he or she feels must get, "as per the industry standards". However, one mustn't also under estimate and quote a lower salary, as it can backfire and can be alost opportunity at the beginning of the career.

Key Management Models, Every Models of Business management explained

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Full notes on extraction of copper and zinc

Extraction of Copper from Copper pyrite

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The principle ore of copper is copper pyrite and copper is extracted from this ore. The different steps in extraction of copper are:

Crushing and Concentration:

The ore obtained from mines are broken down into small piece by jaw crusher and then pulverized. The ore being sulphide ore is concentrated by froth floatation process. Pulverized ore is kept in water containing pine oil and the mixture is agitated by passing compressed air. Ore forms froth with pine oil and comes to the surface and is skimmed off while impurities are left in water.

Roasting

The concentrated ore is heated in excess supply of air on the hearth of reberberatory furnace below its melting point. The different changes during roasting are:

a. Moisture and volatile impurities are driven out.

b. Non-metallic impurities like sulphur, phosphorous, arsenic etc. are removed as their oxides.

S + O2      →             SO2

P4 + SO2        →       2P2O5

c. Copper pyrite dissociates into individual sulphide

2CuFeS2 + O2    →         Cu2S + 2FeS + SO2     major r × n

Small amount of metal sulphides are oxidized to oxides.

2FeS + 3O2     →       2FeO   + SO2             minor reaction

2Cu2S + 3O2   →       2CuO  + 2SO2

Smelting

The roasted ore, coke and silica (flux) is charged a water jacketed blast furnace when hot air is passed into blast furnace. Fes if oxidized to FeO which combines with S1O2 to form ferrous silicate as slag.

                        2FeS + 3O2      →     2FeO + 2SO2

                        FeO + S1O2    →       FeS1O3

                                    Flux                             Ferrous Silicate

As long as Fes is present in the mixture Cu2O can’t be formed as copper has higher affinity for sulphur than oxygen. In molten state FeS & Cu2S are missible and the molten mixture of Cu2S and FeS is called copper malte. The lower end of blast furnace has two openings for slag and copper matte.

Fig: Blast Furnace for extraction of copper

Bassemerisation

The molten matte is mixed with little silica and charged into a Bessemer converter. Lined internally by basic lining of CaO or MgO. Hot air is blown into the mixture which converts remaining FeS. To FeSiO3.

                        2FeS + 3O2     →     2FeO + 2SO2

                        FeO + SiO2     →   FeSiO3 (slag)

The slag is removed and air is passed. The air oxidized Cu2S partially to Cu2O which then reduces remaining Cu2S to metallic copper. During this process no external reducing agent is used so it is an example of self-reduction.

                        2CuS + 3O2   →    2Cu2O + 2SO2

                        2Cu2O + Cu2S    →   6Cu + SO2

The reaction is highly exothermic and copper obtained is in molten state. During solidification, SO2 escapes forming blisters on the surface of metal. This variety of copper containing about 2% of impurity is blister copper.

Refining:

Blister copper consists of about 2% of impurities consisting of cliver, Glod, Zinc, Nicket etc. It is mostly purified by electrolytic method. A block of impure copper is anode, a strip of pure copper is cathode while solution of CuSO4 containing dil H2SO4 is e;ectrolyte. On passing current, impure copper dissolves and equivalent amount of pure copper is deposited at cathode. Impurities are collected below anode as anode mud.

             CuSO4 (aq)      →      Cu++ + SO–4

            At cathode: Cu++ + 2e   →     Cu

            At anode:        Cu         →      Cu++ + 2e

Physical Properties:

• It is a transition metal having characteristic red color.
• It is highly malleable & ductile and has high electrical and thermal conductivity
• It has high melting point 10830c and bpt 2320c
• It has specific gravity 8.93

Chemical properties

Action of air:

Dry air has no effect but moist air containing CO2gas forms a green layer of basic copper carbonate.

Cu + O2+ CO2 + H2O     →  CuCO3Cu(OH)2

                                Moist air                                  basic copper carbonate

On heating in air 9000c, copper forms cupric oxide or black oxide of copper,

4Cu + O2         900 →               2CuO

Cupric oxide

On further heating above 11000c, copper gives cuprous oxide or red oxide of copper.

4Cu + O2         1000 →            2Cu2O

Cuprous oxide

Action of water:

Water has no effect on copper.

Action of Alkalis:

Alkalis has no effect on copper.

Action of Acids:

a) With HNO3

dil HNO3 does not react with copper while moderately conc. HNO3 (1:1) is reduced  to Nitric oxide by copper.

3Cu + 8HNO3              →           3Cu (NO3)2 + 2NO + 4H2O

moderately conc.

Conc. HNO3 is reduced to Nitric oxide by copper.

Cu + 4HNO3              →             Cu (NO3)2 + 2NO2 + 2H2O

Conc

b) With H2SO4 :

dil. H2SO4  alone does not react but hot dil. H2SO4  in presence of air gives CuSO4

2Cu + 2H2SO4 + O2                 →           2CuSO4+2H2O

Hot & dil.

With hot and conc. H2So4, Coppe gives So2 gas.

Cu + 2H2SO4            →              CuSO4 +SO2+2H2O

Hot & conc.

c) With HCl:-

Copper reacts with hot and conc HCl in presence of air forming cupric chloride.

2Cu+4HCl+ O2                   →                2CuCl2+2H2O

5) Displacement reaction:-

Copper can displace metals lying below it in electrochemical series from their salt solution.

Cu+2AgNo3              →              Cu (No3)2+2Ag

3Cu+2Aucl3                 →           3Cucl2+2Au

Uses of Copper:-

i) It is in making electrical cables.

ii) It is used in making coins.

iii) It is used in making allays like brass, German Silver.

iv) It is Rold Gold, constantan, bell metal etc.

iv) It is used in making utensits.

v) It is used in making scientifi equipments like calioriemeter boilers .

Extraction of Zinc from Zinc Blende

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Concentration:

Zinc blende is concentrated by floath floatation process. The pulverized ore is kept in large tank containing water and pine oil. The mixture is agitated by passing compressed air. Ore forms froth and comes to the surface while impurities are left in water.

Roasting:

The concentrated are is heated in excess supply of air above 9000c on the hearth of reberveratory furnace. During roasting. Zinc sulphide is converted to Zinc Oxide.

2ZnS + 3O2                   →          2ZnO + 2SO2

Small amount of ZnS may be oxidized to ZnSO4 but above 9000c , ZnSO4 decompose forming Zinc Oxide (ZnO)

ZnS + 2O2            →                   ZnSO4

                ZnSO4             above 900   →           2ZnO + 2SO2 + O2

Reduction (Smelting):-

ZnO obtained during roasting is mixed with coke and heated strongly where ZnO is reduced to Zn by carbon.

ZnO + C                 →                 Zn + CO

The reduction is done in vertical refort. In this process roasted are mixed with coke in the ration of 2:1 and small briquets are made. These briquets are fed into vertical report furnance, from the charging door. The report is heated externally by burning produce gas (w+N2) to about 14000c. The vapour of zn is camed to condenser where it condenses to give molten zinc called spelter zinc.

Purification:-

Zinc spelter contains pb, fb, cd, as, etc. as impurities. Impure zinc can be purified by following methods.

a. By fractional distillation:-

The bpt of Pb, Fb are higher than that of zinc while that of cadmium, arsenic are lower than that of zinc. When distillation is carried out around 1000°c, zinc, Cd, As, etc. distill off leaving Pb and Fe the distillate is then heated to 800°c where cd and as distill off leaving pure zinc. This sample of Zn is about 99% pure.

b. By electrolysis:-

Zinc of higher purity can be obtained by electrolysis. Pure zinc rod is used as cathode while a block of impure zinc is used as anode. A mixture of ZnSO4 and dill H2SO4 is used as electrolyte. On passing current impure zinc dissolves and equivalent amount of pure zinc is deposited at cathode.

Physical Properties

i. It is a blueish white lusticous metal.

ii. zinc is brittle at ordinary temperature but it becomes malleable from 100-150. c then again it become brittle.

iii. It melts at 420. c and boils at 900. c and has sp. Gravity 7.13.

Chemical Properties

1. Action of air:-

Dry air has effect on zinc but in moist air, zinc forms a protective layer of basic zinc carbonate.

Zn + O2 + H2O + CO2         →                           ZnCO3.Zn (OH)2

When heated in air, zinc burns greenish blue flame forming clouds of light, white power of zinc oxide which is commonly known as ‘philosopher’s wool. Beside this name, other common names are ‘Zinc white and china white.

2Zn + O2      →               2ZnO

2. Action of water:-

Pure zinc does not react with water but impure zinc displaces hydrogen gas. Zn-inc reacts with hno3 in four different concentrations giving different reduced product.

Zn + H2O          →          ZnO + H2

3. Action with acids:

Dilute H2SO4 and dil. HCl gives H2 gas with zinc

Zn + dil. H2SO4         →             ZnCl2 + H2

Zn + dil. HCl                →          ZnSO4 + H2

With hot and conc. H2SO4 zinc gives SO2 gas

Zn + 2H2SO4              →             ZnSO4 + SO2 + 2H2O

Zinc reacts with HNO3 in four different conditions giving different product:

i. with very dil. HNO3

With very dil. HNO3, Zn gives ammonium nitrate

4Zn + 10HNO3            →           4Zn(NO3)2 + NH4NO2 +3H2O

ii. with dil. HNO3

Zinc reduces dil. HNO3 to nitrous oxide (N2O)

4Zn + 10HNO3                  →     4Zn(NO3)2 + N2O +5H2O

iii. With modrately conc. HNO3

Zinc reduces moderately conc. HNO3 to NO (nitric oxide)

3Zn + 8HNO3               →          3Zn(NO3)2 + 2NO2 +4H2O

iv. With conc. HNO3

Zinc reduces conc. HNO3 to NO2 (nitrogen dioxide)

Zn + 4HNO3                →           Zn(NO3)2 +2 NO2 +2H2O

4. Reaction with Alkalies:-

Zinc readily dissolve in hot and conc. solution of caustic bases like NaOH , KOH, etc giving hydrogen gas.

Zn + NaOH             →                Na2ZnO2 + H2

Zn + KOH                  →            K2ZnO2 + H2

5. Displacement Reaction:-

Zinc can displace less electropositive metals from their salt solution

Zn + CuSO4                     →      ZnSO4 + Cu

Zn + 2Na[Au(CN)2]           →                 NA2[Zn(CN)4] + 2Au

Uses of zinc:

- It is used for galvanization of iron.

- It is used in making alloys like Brass. German metal, German  silver, etc

Galvanization

The process of applying a coat of zinc on base metal like iron is galvanization Iron is galvanized to protect iron from resting the galvanization is done in following steps.

i. cleaning and picking:

Before applying zinc coat on iron, the surface of iron should be clean. The cleaning is done first by sand blast and then washed by dipping in diluted acid. The process of cleaning iron by using dil. acid is called picking.

ii. Applying of zinc coat:

The coat of zinc can be applied on iron surface by 3 methods.

a. By electroplating:-

Zinc can be electroplated on iron by keeping iron article as cathode, zinc as anode and znso4 solution as electrolyte. On passing current a layer of zinc is coated on iron article.

b. By metallizing :-

Zinc can be directly applied by dipping iron article in molten zinc this process is generally used for galvanization of iron sheets. Iron sheets are dipped in molten zinc bath & then passed through huge rollers that distributer zinc coat equally.

c. By sherardizing:-

The process is used for galvanization of small iron articles like nail, screws etc. In this process iron article are mixed with zinc dust and then heated in enclosed vessel for few hours when coat of zinc is applied on iron.

·         It is used in making amalgam.