# Perfect Numbers

From Lee Sallows:

As the reader can check, the English number names less than “twenty” are composed using 16 different letters of the alphabet. We assign a distinct integral value to each of these as follows:

```E   F   G   H   I   L   N   O   R   S   T   U   V   W   X   Z
3   9   6   1  -4   0   5  -7  -6  -1   2   8  -3   7  11  10
```

The result is the following run of so called “perfect” numbers:

```Z+E+R+O       =   10 + 3 – 6 – 7          =    0
O+N+E         =   –7 + 5 + 3              =    1
T+W+O         =    2 + 7 – 7              =    2
T+H+R+E+E     =    2 + 1 – 6 + 3 + 3      =    3
F+O+U+R       =    9 – 7 + 8 – 6          =    4
F+I+V+E       =    9 – 4 – 3 + 3          =    5
S+I+X         =   –1 – 4 + 11             =    6
S+E+V+E+N     =   –1 + 3 – 3 + 3 + 5      =    7
E+I+G+H+T     =    3 – 4 + 6 + 1 + 2      =    8
N+I+N+E       =    5 – 4 + 5 + 3          =    9
T+E+N         =    2 + 3 + 5              =   10
E+L+E+V+E+N   =    3 + 0 + 3 – 3 + 3 + 5  =   11
T+W+E+L+V+E   =    2 + 7 + 3 + 0 – 3 + 3  =   12
```

The above is due to a computer program in which nested Do-loops try out all possible values in systematically incremented steps. The above solution is one of two sets coming in second place to the minimal (lowest set of values) solution seen here:

``` E   F   G   H   I   L   N   O   R   S   T   U   V   W   X   Z
–2  –6   0  –7   7   9   2   1   4   3  10   5   6  –9  –4  –3
```

But why does the list above stop at twelve? Given that 3 + 10 = 13, and assuming that THREE, TEN and THIRTEEN are all perfect, we have T+H+I+R+T+E+E+N = T+H+R+E+E + T+E+N. But cancelling common letters on both sides of this equation yields E = I, which is to say E and I must share the same value, contrary to our requirement above that the letters be assigned distinct values. Thus, irrespective of letter values selected, if it includes THREE and TEN, no unbroken run of perfect numbers can exceed TWELVE. This might be decribed as a formal proof that THIRTEEN is unlucky.

But not all situations call for an unbroken series of perfect numbers. Sixteen distinct numbers occur in the following, eight positive, eight negative. This lends itself to display on a checkerboard:

Choose any number on the board. Call out the letters that spell its name, adding up their associated numbers when on white squares, subtracting when on black. Their sum is the number you selected.

(Thanks, Lee.)

# An Alphageomagic Square

From Lee Sallows:

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# Ordnance

From Lee Sallows:

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# Measure for Measure

From Lee Sallows:

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# Inventory Control

In 2015 British computer scientist Chris Patuzzo produced a self-enumerating pangram — a sentence that itemizes its own contents — that records its totals as percentages:

This sentence is dedicated to Lee Sallows and to within one decimal place four point five percent of the letters in this sentence are a’s, zero point one percent are b’s, four point three percent are c’s, zero point nine percent are d’s, twenty point one percent are e’s, one point five percent are f’s, zero point four percent are g’s, one point five percent are h’s, six point eight percent are i’s, zero point one percent are j’s, zero point one percent are k’s, one point one percent are l’s, zero point three percent are m’s, twelve point one percent are n’s, eight point one percent are o’s, seven point three percent are p’s, zero point one percent are q’s, nine point nine percent are r’s, five point six percent are s’s, nine point nine percent are t’s, zero point seven percent are u’s, one point four percent are v’s, zero point seven percent are w’s, zero point five percent are x’s, zero point three percent are y’s and one point six percent are z’s.

The next challenge was to extend the precision beyond one decimal place. Impressively, Matthias Belz produced this specimen in 2017:

Rounded to five decimal places, two point six five two five two percent of the letters of this sentence are a’s, zero point zero eight eight four two percent are b’s, two point six five two five two percent are c’s, zero point four four two zero nine percent are d’s, nineteen point eight zero five four eight percent are e’s, three point four four eight two eight percent are f’s, one point seven six eight three five percent are g’s, two point nine one seven seven seven percent are h’s, seven point eight six nine one four percent are i’s, zero point zero eight eight four two percent are j’s, zero point zero eight eight four two percent are k’s, zero point three five three six seven percent are l’s, zero point one seven six eight three percent are m’s, ten point two five six four one percent are n’s, eight point nine three zero one five percent are o’s, four point seven seven four five four percent are p’s, zero point zero eight eight four two percent are q’s, nine point five four nine zero seven percent are r’s, four point nine five one three seven percent are s’s, nine point six three seven four nine percent are t’s, two point zero three three six zero percent are u’s, two point seven four zero nine four percent are v’s, one point six seven nine nine three percent are w’s, zero point nine seven two five nine percent are x’s, zero point zero eight eight four two percent are y’s and one point nine four five one eight percent are z’s.

These numbers are still rounded, so later that year he surpassed that with an instance giving precisely accurate values:

Exactly three point eight seven five percent of the letters of this autogram are a’s, zero point one two five percent are b’s, three point five percent are c’s, zero point two five percent are d’s, twenty-one point two five percent are e’s, three point seven five percent are f’s, zero point three seven five percent are g’s, one point five percent are h’s, seven point two five percent are i’s, zero point one two five percent are j’s, zero point one two five percent are k’s, zero point three seven five percent are l’s, zero point two five percent are m’s, nine point seven five percent are n’s, seven point five percent are o’s, six point five percent are p’s, zero point one two five percent are q’s, nine point three seven five percent are r’s, five point one two five percent are s’s, ten percent are t’s, zero point three seven five percent are u’s, four point six two five percent are v’s, one point five percent are w’s, zero point five percent are x’s, zero point three seven five percent are y’s and one point five percent are z’s.

Details are here.

# Inventory

From Lee Sallows:

(Thanks, Lee!)

# A Polyhex Square

From Lee Sallows:

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# Constitutional Crisis

From Lee Sallows:

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# Fair and Square

From Lee Sallows:

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# Eight Flights

From Lee Sallows, a geomagic square with a staircase theme:

(Thanks, Lee!)