When 11 is binary and 2 plus 1 is denary?
Nope (however you might be right but it’s not the answer I was looking for).
Let’s try …
11 + 2 = 13
13 = ‘A Baker’s Dozen’ ie ‘One’
It beat me.
I think Mrs R got it though. 11 am plus 2 hours is 1 pm.
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Correct.
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I misread the question 
I thought it said when does 11 equal 2 plus 1. Sorry!
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Ages and Ages …
Allan is twice as old as Bill, whereas Bill’s age is one third of Charlie’s age. Altogether their ages add up to 72.
How old are Allan, Bill and Charlie ?
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Bill is 12, Allan is 24, and Charlie is 36.
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Spot-on, as usual Mike. Well done !
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Seven Year Old Grandson !!!
My 7 year old grandson called in this afternoon on his way home at the end of the week from school. (he was with his 5 year old brother and of course his mum).
Mrs D likes them to outline what they have been doing the past week at school. Today, I was casually listening when suddenly my ears pricked up when the 7 year old said “… the sum of any three consecutive numbers is divisible by three …”
…is he correct, or not ?
I think that’s correct. My reasoning as follows.
The sum of any random consecutive sequence can be summarised as:
n + (n+1) + (n+2)
= 3n+3
This can always be divided by three:
(3n+3)/3 = n+1
So choose any random number, n, as your start point. Add the next two consecutive numbers then divide by three and you get your original number +1.
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Well done Bobby - spot-on !
And your explanation has saved me from posting mine - they are effectively the same.
The only slight difference was to note that 3n + 3 = 3(n+1) which is divisible by 3, leaving n+1.
ie effectively the same as yours.
Well done.
This Seven Year Old Grandson strikes again !!!
By way of a follow-up, our young grandson added “… the sum of any two consecutive odd numbers is a multiple of 4 …”
So, again, is he correct, or not ?
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The sum is n + (n +2) = 2n + 2
As n is an odd number, the sum is always divisible by 4.
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As usual Mike. spot-on again. Well done !
I’ll post my (similar) solution below - it has a few extra lines 
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Even numbers can be expressed as 2n
Odd numbers can be expressed as 2n+1
(Where n is an integer).
Two consecutive odd numbers can be expressed as (2n+1) and (2n+3)
So, adding them together gives :-
(2n+1) + (2n+3)
= 4n+4
= 4(n+1)
Dividing by 4 = n+1
n+1 is a whole number, (remember n is an integer !)
Hence the sum of two consecutive odd numbers is always a multiple of 4.
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My Seven-Year-Old Grandson has been at it again !
Following on from “the sum of any two consecutive odd numbers must be a multiple of four” his next announcement was “The sum of the squares of any two consecutive odd numbers cannot be a multiple of four !”
Is he right yet again ?
BTW, I think he’s been talking to his 15 year old brother
You can of course, always try a few examples to see if there is consistency one way or t’other, but unless you try every pair of consecutive odd numbers …
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Well, I think he is right.
Two consecutive odd numbers can be expressed as (2n+1) and (2n+3), therefore the sum of the squares of any two consecutive numbers is:
(2n+1)(2n+1) + (2n+3)(2n+3):
(4nn+2+4n) + (4nn+9+12n) = 8nn + 16n +11
Which, when divided by 4 becomes:
2nn + 4n + 11/4
As 11/4 = 2.75 any sum of the squares of any two consecutive odd numbers will always end with the decimal .75, and cannot be a multiple of 4.
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Mike, your method looks perfect and well explained but has a slight error in the application of it. The first expansion should be (4nn+1+4n). Check: if n=0 the answer should be 1+9=10, not 11.
My approach is similar, though I start from (2n-1) and (2n+1) as conveniently this allows some terms to cancel. It leads to 8nn + 2 which is not divisible by 4.
@Don has some smart grandsons!
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Oranges and Lemons
It was a nice warm, sunny afternoon today so we sat outside sipping a mixture of orange juice and lemon juice whilst we chatted.
I filled a glass half full of orange juice and another glass, twice the size, one-fifth full of orange juice.
I then topped up both glasses with lemon juice and poured the contents of both glasses into a large jug.
How much of the mixture in the jug is lemon juice and how much of it is orange juice ?
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