"A Puzzle A Day" - Flowers, and Venn Diagrams

The next puzzle can be solved longhand, but the best tool I would recommend is a Venn diagram, which solves the puzzle and elegantly shows your working (which my maths teachers said was always a good thing).

"Betty was making paper flowers for the local carnival.  11 flowers had red in them, seven had yellow in them, and five had red and yellow in them.  How many flowers did she make?"

The short answer (11+7+5 = 23) is incorrect; the point here is that the 11 flowers with red in them includes the five that had both red and yellow in them.  Similarly, the seven yellow flowers includes the five that had red in as well.

The quickest way to solve this is with a Venn diagram - see below.  Each circle contains the number of flowers of that colour, and the overlap shows the number of flowers which contain both red and yellow.

We know that there are five flowers which have both colours in them, so we can write 5 in the intersection (overlap) point of the two circles.  We also know that the red circle in total must contain 11 flowers, so the area outside the overlapping section is 11-5.  Finally, for the yellow the circle, the total needs be 7.

Simplifying this, we can see that there are 6 flowers that are red-only, 2 that are yellow-only, and five that are both red and yellow:

And hence the total is 13.  The initial "23" (11+7+5) counts the overlapped area twice more than it should be (i.e. it's counted as a red flower, a yellow flower and as a red-and-yellow flower).  By drawing out a simple Venn diagram, it's easy to see what the correct solution is.

Other articles in the 'A Puzzle A Day' series:

Three Horse Race
How Old is Aunt Tabitha
Big Ben Strikes Twelve

Around Goes The Gossip
Sugar Cubes (Orders of Magnitude)
Adding Nine, Reversing the Digits
Flowers and Venn Diagrams
Sum of Three Numbers equals Their Product

Other Calculator Fun and Games articles:

Snakes and Ladders (Introducing the Collatz Conjecture)
Over and Out (reduce six-digit numbers to zero as rapidly as possible)
Calculator Games: Front to Back
Calculator Games: Investigating the Kaprekar Constant

"A Puzzle A Day" - add nine, reverse digits

This puzzle is one of those which is almost as simple to solve as it is to state:

Find a two-digit number that reverses its digits when you add nine to it.

Knowing that the nine-times table contains numbers which have two series of ascending tens and descending units, this should be a case of just identifying a pair:

9, 18, 27, 36, 45, 54, 63, etc.

However, there are many more pairs of numbers that fit the requirement in the question:

23 and 32
34 and 43
45 and 54 we've already mentioned
56 and 65
and so on

Any two digit number of the form (10x + (x+1)) or 11x+1 will reverse its digits when adding nine (up to 89 + 9 = 98).

The situation changes above 100, and its not possible to reverse digits by adding just nine (and we have three-digit numbers).

If you like this article, can I recommend this one:  Calculator Games: Front to Back?  It's on a similar theme.

Other articles in the 'A Puzzle A Day' series:

Three Horse Race
How Old is Aunt Tabitha
Big Ben Strikes Twelve

Round Goes The Gossip
Sugar Cubes (Orders of Magnitude)
Add Nine, Reverse Digits
Flowers and Venn Diagrams
Sum of Three Numbers Equals Their Product

Other Calculator Fun and Games articles:

Snakes and Ladders (Collatz Conjecture)
Crafty Calculator Calculations (numerical anagrams with five digits)
More Multiplications (numerical anagrams, four digits)
Over and Out (reduce large numbers to zero as rapidly as possible)
Calculator Games: Front to Back
Calculator Games: Up, up and away with Ulam sequences
Calculator Games: The Kaprekar Constant

"A Puzzle A Day" - How Old Is Aunt Tabitha (age in days)

Next puzzle - another interesting one: 

Aunt Tabitha was extremely touch about her age.  When an impudent nephew was brave enough to ask her, she cunningly replied that she was 35 years old, not counting Saturdays or Sundays.  How old was she?

So Aunt Tabitha has given her age in weekdays, and weekdays only account for five-sevenths of the total week.  Therefore, Aunt Tabitha's stated age of 35 is only five-sevenths of her actual age; 35 / (5/7) = 49.

So she's 49 years old.  But I'll leave it to the impudent nephew tell her that.

It might have been more interesting if she'd given her age as 14 years old in weekends - which is also a more exciting and appealing way of describing her age.

Having said all that, I can say that I'm 30 years old (excluding weekends), or, even better, 12 years old in weekends.  I'm not touchy about my age, but the thought of being 12 years old in weekends (and not even a teenager) is definitely more appealing than stating my actual age!

If you've enjoyed this puzzle and found this post useful, you may also be interested in some of my other posts on a similar theme:

Snakes and Ladders (Collatz Conjecture)
Crafty Calculator Calculations (numerical anagrams with five digits)
More Multiplications (numerical anagrams, four digits)
Over and Out (reduce large numbers to zero as rapidly as possible)
Calculator Games: Front to Back
Calculator Games: Up, up and away with Ulam sequences
Calculator Games: The Kaprekar Constant

"A Puzzle A Day" - A Three Horse Race

This puzzle comes from the Puzzle A Day pad (it's meant to last a year; at this rate I'll be completing it in 2025).

The question is simple (but with a slight twist):  "How many ways can a three horse race finish, including ties?"

The simple answer (excluding ties) can be found by looking at all the possible combinations; if we call the horses A, B and C, then we have:

ABC
ACB
BAC
BCA
CAB
CBA

Six ways (3*2*1 = 6, for those who are considering combinations and permutations).

Now, refreshing the list and including the ties (which will be shown in brackets):

ABC
A(BC)
(AB)C
ACB
A(CB)
(AC)B
BAC
B(AC)
(BA)C
BCA
B(CA)

(BC)A

CAB
(CA)B
C(AB)

CBA
(CB)A
C(BA)
(CAB)

13 in total - there are three variations for each of the previous combinations, so ABC now becomes ABC, (AB)C and A(BC) - but we must now deduplicate A(BC) and A (CB), and so on.  The red text above shows a duplicate of a combination which has already been seen above it in the list.

And there's one where all three tie together (ABC).  The need to deduplicate the ties makes this question more complex than it appears at first sight, and so (as is often the case) care is taken to fully understand the question.

Other articles in the 'A Puzzle A Day' series:

Three Horse Race
How Old is Aunt Tabitha
Big Ben Strikes Twelve

Round Goes The Gossip
Sugar Cubes (Orders of Magnitude)
Add Nine, Reverse Digits
Flowers and Venn Diagrams
Sum of Three Numbers Equals Their Product

Another random puzzle:

Over and Out (reduce a six-digit number to zero in six arithmetic steps)

"A Puzzle A Day" inspired by Prof Rubik: Big Ben Strikes Twelve

I've recently bought a "Puzzle a Day" - it's a block of 365 single-sheet puzzles (intended to be one a day for a year) on a charity shop visit.  It was new and unopened (I should have taken the hint and walked away immediately), but I thought it might stretch my brain in new ways, and some of them will probably be blog-worthy.

Some are, some aren't.

The first couple don't translate well to a blog article, but Puzzle 3 is interesting: 

"If it takes Big Ben six seconds to strike six o'clock, how long will it take to strike twelve o'clock?"

The immediate (and wrong) answer is twelve seconds; the trick is this:  there is no time to be measured after the sixth ring.  The duration of the rings is not one second between the first and second, then another between the 2nd and 3rd, 3rd and 4th, 4th and 5th, 5th and sixth, and then a second after the sixth ring.  In fact, there are six fifths (1.2 seconds) between each chime. The sixth chime occurs after the previous five have rung out, = 5 * 1.2 seconds = 6 seconds.

Now that we know that there are 1.2 seconds between each chime, we need to calculate the length of 11 chimes (knowing that the 12th chime will occur immediately afterwards).

11 * 1.2 = 13.2 seconds, or 13 1/5 seconds.

Not 12 seconds (as you may have immediately guessed).

Sorry ;-)

Other articles in the 'A Puzzle A Day' series:

Three Horse Race
How Old is Aunt Tabitha
Big Ben Strikes Twelve

Round Goes The Gossip
Sugar Cubes (Orders of Magnitude)
Add Nine, Reverse Digits
Flowers and Venn Diagrams
Sum of Three Numbers Equals Their Product

Another random puzzle:

Front to back (palindromes in numbers)

Transformers Trading Card Game: Data Analysis

After my previous post, where I started playing the Transformers Trading Card Game, I started looking at the types of battle cards and doing some back-of-the-envelope data analysis.  I don't expect to find any trends, but I wanted to take a look at how the battle cards break down by type (colour, and action vs upgrade, and type of upgrade).

In my previous post, I carried out some quick high-level analysis: how many of each type of card are there?

I was surprised to see that more than half of the cards are in the Action group, while less than one in eight is Armour.  It certainly made me pay more attention to the cards I was choosing for my deck, and to consider the types of cards I was using (and not just if they would be fun to play - in theory).

Let's break them down further - what colour icons do the cards have?  White; blue; orange or green?

I know tables aren't to everybody's liking, but here's a quick, simple table that breaks down the card colours and types:

Some thoughts (from a data perspective)

- There's an approximately equal quantity of pure blue cards and pure orange cards (46 and 48); twice as many as pure white (24).  
-  White is only combined with green, not with blue or orange.  That makes sense from a game-play perspective; when attacking or defending, it would be too much to get the benefit of an orange (or a blue) and then also draw two extra cards to further add to your total.
- There are three cards which are blue/orange (an interesting and useful combination in the game); these are Recon System (which is a star card); Matrix of Leadership (can be used only on Autobots) and Roll Out!

 

Now, these are all great cards since they provide both defence and offence boosts.

- It is possible, but not necessarily desired, to play all white; all blue; all orange; or all blank (no icons) - any row that sums to 14 or more will produce a complete deck of that type.  There has been a lot of play with all-blue (or mostly blue) 'defensive' decks, and all or mostly orange "aggressive" or "aggro" decks.  My personal preference is towards a mixed deck with a lot of white, but I'm still working on it (I'm looking to play Mirage; Hound and Jazz, and other 'white-friendly' teams).

Wave 3 of the Transformers TCG is coming up in less than two weeks, so I'll probably be doing further analysis on how the overall card profile changes.  I'm personally looking to get the new Red Alert (with his liking for mixed decks) and Private Stakeout (who likes white), and to play them with Hound and Jazz; or Mirage/Bumblebee BW/Jazz/Private Stakeout.  We'll see!

Transformers Trading Card Game: A Newbie's View

I have long been a fan of Transformers.  I was in the target audience age of 6-10 when the first comics came out, and my first issue was issue 4, back in November 1984.  I've reviewed Dark of the Moon, and Revenge of the Fallen.  I was delighted (but mostly interested and curious) when I learned of the Transformers Trading Card Game a few months ago, and after watching quite a few YouTube videos, decided to take the plunge and buy some of the cards on eBay.  It's all relative, but £4 for a pack of cards seems expensive, while buying one or two on eBay seemed less so. 

I was impressed with the size, colouring and designs (very reminiscent of G1), so I bought a starter pack (Wave 1 Autobots), and tried my first player-versus-player match with the starter pack cards (I played both players), working through the game mechanics, keeping track of damage with a notepad and pencil, and observing while playing.  I'm no expert at card games (this is my first, and will probably be my only), but I started to notice a few issues with the way I was playing.

A brief summary of my first game: 

Decepticons:  Flamewar, Starscream (Scheming Second in Command), Megatron (Decepticon Leader)

Autobots:
Autobot Hound; Optimus Prime (Autobot Leader); Bumblebee (Courageous Scout)

The Autobots took a pounding due to Megatron's pierce ability; Optimus and Bumblebee were KO'd with just two turns.  Autobot Hound lasted a little longer due to his better health and I was able to repair damage from him a couple of times.  It was a narrow Autobot victory.

In between games 1 and 2, I purchased an additional 58 battle cards - a mix so that I can understand what style suits me, and what I need to play.  I also purchased the Stunticons, so I make sure I have the relevant Stunticon cards to form Menasor.

Game two
Changes from previous game: 

Decepticons 
Exchange Starscream (SSIC) for Ramjet;

Autobots change to Dinobot Swoop (Fearsome Flyer), Optimus Prime (Freedom Fighter)and Optimus Prime (Autobot Leader). 

I modify the decks to make the Decepticon deck more aggressive, while the Autobot deck includes cards like Fling, Field Repair and Repair Bay so that I can last longer in battles (in theory).  At this point I'm still not overly worried about the colours of the icons, I'm just selecting battle cards based on their abilities.

There were a number of issues I had while playing the game here:
- I was getting handfuls of action cards and almost no upgrades.  The action cards enabling me to "Play an upgrade" were useless, and the ability to play an upgrade each turn was being wasted
 - The other hands I were playing ran out of cards.  I wasn't drawing enough  - and this makes sense:  if I can only draw a card, but play an action and an upgrade, I was going to be decreasing my hand by one card per turn until I ran out.  I needed more draw capability.

I did some analysis of the overall cards in Waves 1 and Wave 2, just as a brief diversion:  There are 172 battle cards in total, of which some are duplicates.  I'll produce a deduplicated list another time:
90 (over half) are Action cards
31 are Utilities (and I was not putting enough of these in my decks)
30 are Weapons
21 are Armour

Just based on random sampling, it was no wonder that I was running out of upgrade cards (with so many action cards to choose), and then running out of cards altogether.  I had to re-balance my decks to include more upgrades and fewer actions.

And I needed more draw capability.

Game three
Decepticon PowerHouse versus Mixed Drawers

Decepticons
Megatron (Decepticon Leader)
Ramjet (Sky Smasher) - who needs abilities when you have 7 attack and 14 health?
Flamewar  - a trusty sidekick who brings the star count up to 25

Mixed Drawers
Autobot Hound (Long Range Scout) - when you flit to alt mode, possibility to scrap one and draw two
Optimus Prime (Autobot Leader) - when you flit to alt mode, draw a card
Dead End - when you flip to bot mode, draw a card
Decepticon Drag Strip - brings the total to 25 stars; and when attacking in alt mode and you get a blue and orange icon, draw 2 cards.

Results
Better.  Ramjet and Megatron are still lethal, especially when given upgrades (Ramjet especially, as he has no abilities of his own).  I drew a "Disarm" for the Mixed Drawers, which ended Ramjet's resistance, and they eventually won.  I like that card!

I know I'm probably playing a biased game, but it's not completely deliberate.  I'm just subconsciously rooting for Optimus and Hound (my favourite G1 characters).

Questions

However, all this single-handed playing has led me to ask some questions:

1. How do you (as a player) select your characters and battle cards?  Do you pick characters with cool abilities, and then build a deck around them?  Or do you look at the battle cards and think "That would be fun, if I could find the right mêlée/ranged/leader character to play it with"?

2.  How do you balance the proportion of actions/armour/utilities/weapons?  What split do you use?

3.  How do you increase your draw capacity?  Cards, or flip abilities?

My ongoing battle card deck

This is still a work in progress, but here's what I'm going to try in my next game

Actions
Inspiring Leadership x3
Repair Bay x3  (actually, I only own 2 at the moment... need to fix this!)
Rest and Relaxation x3 (I know it's a green icon, but that much repair is very useful)
Fling x2
Field Repair x2
Treasure Hunt x2 (still upping my upgrade count)
Disarm x2 (this was a killer against Ramjet and I liked playing it)

Armour
Evasive Maneuvers x2
Body Armor x2
Scrapper Gauntlets x3
Reinforced Plating x2
Cooling Vents x2

Weapons
Grenade Launcher x2
Handheld Blaster x2
Enforcement Batons x2

Utilities
Debilitating Crystal x2
Security Console x2
Data Pad x2

17 Actions
11 Armour
6 Weapons
6 Utilities
40 total.

Any thoughts?  Next time - a data analysis view of the Transformers Card Game, looking at the different coloured pips on the cards.