When I was about 10 years old, long before internet, I chanced upon a book whose name I do not recall, and whose contents were math riddles. I guess I can call it easily my first love, and apologies to the lady who knows she was the first. Well, that lady was my first human love, ok?

One riddle stuck in my head, and it went something like this: using four times the number 4 and all possible mathematical operations, operators and notations, create all the numbers 0 to 10. You know, something like:

0 = 4–4+4–4

1 was also easy, 5 was a bit more complicated... I did it.

Forty years later, while starting a writing project which was a personal memoirs book, it dawned upon me that I would better make it even more interesting than it (really) was by sprinkling a math quiz of my own throughout it, which will deal with four times the number 5. And go all the way from 0 to... as far as possible, sequentially of course. I had no idea what an exciting minefield I was getting into, and once I finally got through all the way and... further, what a pleasure would the accomplishment of this quiz in its entirety bring into my life.

Ten years further, and the memoirs book was published, quiz included, and ten years even further on I developed it... even further. Resulting in a re-publishing of afore said book and in this book, which is dedicated entirely and only to the quiz (thus, saving you the cost of buying the memoirs book in case all you want is to face the challenge of this lengthy quiz).

OK, now the rules which will do their best to limit you, guide you, and eventually... misguide you. I define my own rules, to keep my distance from other, similar quizes:

1. All “high school” accepted mathematic (arithmetic, trigonometry, calculus, etc.) operations, signs and symbols are allowed. I will refrain from listing them all, however there are some “exotic” ones out there in the games’ world, and I will tell you those I specifically do NOT allow (some of which you had never heard of anyway): subfactorial, superfactorial, double factorial, overline, decimal point. (The last two are popular when creating fractions, not in my schoolyard, sorry.) This will also keep you from trying to find solutions on the internet.

2. ALL four 5’s must be used.

3. If you find a way to use physics, chemistry, biology, thermodynamics, psychology and poetry to help out, though I wouldn’t really know how – feel free. Even if, honestly, I doubt that you need it.

4. Calculations are in the normal basis-10 numbers domain only.

5. Operations resulting in positive and negative answers (for example: the 4th root of 16 = ±2) ONLY the positive answer (+2) is retained.

6. Boolean functions are logical, not mathematical functions, therefore they are excluded: Yes, everyone knows, for example, that in the boolean world:

1+1 = 1

however all “boolean persons” know as well that above “+” means actually “OR” and is not a mathematical function. Similar with any other boolean or programming functions (like INT for INTeger, etc).

7. No mix of normal (so called Arabic) numerals with other types of numerals – Roman, Sanskrit, Hebrew, Hieroglyphs and similar.

8. It is not allowed to use symbols representing well defined numbers [like in mathematics π(=3.14...), e(=2.71...), i(=√-1), %(=1/100), ∞(=infinity), etc. or like in physics c (the speed of light in vacuum), h (Planck’s constant), G (the gravitational constant), etc.]. Otherwise, any natural number could be easily achieved, like for example:

1 = e/e

2 = (e+e)/e

and so on and so forth.

Please, also do not use also ∏ with the meaning of “product” of family members, same as ∑ with the meaning of “summation” of family members (reasons for it in the “excuses” chapter) however you can use ( ) “binomial coefficient” or Γ “gamma” and similar if, for whatever strange reason, you find it helpful.

9. Cheap tricks, though I cannot define all of them, are excluded as well. A matter of ethics. Like for example including expressions type (X representing here the mathematically accepted symbol for “variable”):

X/X

to get an extra 1 when missing it. Beyond being incorrect (since X could also be 0 where above calculation would be, mathematically speaking, illegal) this is also, as said, cheap and vulgar and unethical. And actually unnecessary. Even when such a calculation would be close to legal, like in:

X–X

to get a 0 and this 0 could help you to get rid of a number which is kind of leftover, by multiplying them... no, please do not use it. Same applies to using an expression type:

(hydrogen atoms in a water molecule)

when you are missing a 2 to accomplish the feat; this is also, as you well understand, not very mathematical in nature. On the other hand, talking about an expression type:

sin²X+cos²X

or

sin²n+cos²n

to get an extra 1 to help you with your calculation when having two superfluous 2’s to use (I know, makes no sense, just an example) is more borderline. Though perfectly legal and always correct, it would leave an aftertaste because it would leave an undefined variable in the final composition, and I tend to say no, but I leave it to you. X is, of course, the symbol for any number on the X axis, and n is the symbol for any natural number. If instead of X or n you would use a number, say 2, this would be perfectly fine and:

sin²2+cos²2 = 1

would be an elegant way to get a 1 from four 2’s. Nevertheless, there are usually easier and much simpler ways to get the same.

10. If, God knows when, you need to do it, feel free to add/multiply/etc. numbers even if they have different units; e.g. 5kg+5inch=10 is perfectly acceptable - we care only about the numbers involved. To help you and... careful, maybe to confuse you, let's agree that: time is measured in seconds (years would do as well), distance in feet (for Europeans – go for meters), temperature in degrees Fahrenheit (you prefer Celsius? fine with me), angles in degrees (limited 0 to 360), and people in IQ (only above 100 please, quizer’s prerogative).

11. Last but not least and neither mandatory, try to get as simple a result as possible. For example, 0 can be achieved in several ways, all quite simple:

0 = 5-5+5-5

or

0 = (5+5)x(5-5)

or

0 = 5/5-5/5

It would be acceptable and correct also if it was achieved by:

0 = (sin(5-5))x(ln(arctg(5))x5)

but kind of cumbersome, don't you think?

All above is based on remarks I got, misgivings I felt and pure need.

OK, I did my best to define the game, the rules, the playfield, I may have overlooked some points and some ingenious tricks but I hope I didn’t.

So, let’s start? With:

1 = ????

NOTE: The answers are always printed on the even pages of this book, to prevent accidental peek-a-boo spoilers. However, no one’s going to know if in a moment of weakness, you did peek. You’ll just spoil your own pleasure of missed discovery and elation. Be warned: solutions DO exist for all the queries presented, and there are no dirty tricks involved, scout’s word of honor (this, even if I have never been a scout, as you’ll find out later).