In this puzzle, letters have been consistently replaced by digits, where:
• The same letter is substituted by the same digit.
• Different letters are substituted by different digits.
• No number can contain any leading zero.
It is known that each of the numbers FOUR and FIVE is a perfect square.
Furthermore, it is known that the each of the numerical answers to the undernoted queries is constituted entirely by the digits 4 and 5.
• (a) How many squares are strictly between FOUR and FIVE?
• (b) What is one-tenth of TEN - FIVE + FOUR?
• (c) What is the lowest positive integer that must be added to TEN to make it a perfect square?
Determine the value of FORTUNE.
Source: Adapted from Enigma #674 that appeared on "New Scientist" in 1992.
clearvars,clc
i=0;
for n=ceil(sqrt(1000):floor(sqrt(9999)))
i=i+1;
sq(i,:)=char(string(n^2));
end
for a=1:length(sq)
for b=a+1:length(sq)
if isequal(sq(a,:),'1024') && isequal(sq(b,:),'1369')
xxx=9;
end
if sq(a,1)==sq(b,1)
un=union(sq(a,:),sq(b,:));
if length(un)==7
n1=sqrt(str2double(sq(a,:)));
n2=sqrt(str2double(sq(b,:)));
if n2-n1-1==4 || n2-n1-1==5
disp([sq(a,:),' ',sq(b,:)])
disp(n2-n1-1)
this=[sq(a,:),sq(b,:)];
lets='fourfive';
for letter=1:length(lets);
build=[lets(letter) '=' this(letter) ';'];
eval(build);
end
for t=1:9
for n=1:9
used=union(un,char(string(t)));
used=union(used,char(string(n)));
if length(used)==9
ten=100*t+10*e+n;
four=str2double(sq(a,:));
five=str2double(sq(b,:));
disp([(ten-five+four)/10 ten]);
end
end
end
end
end
end
end
end
finds (with my annotations)
bounds possible values of
(TEN-FIVE+FOUR)/10 TEN
1024 1369
4 intervening squares
25.2 597
25.3 598
45 795
45.3 798
55 895
55.2 897
2304 2916
5 intervening squares
-4.5 567
-4.4 568
15.3 765
15.6 768
25.3 865
25.5 867
7056 7921
4 intervening squares
-55.1 314
-54.7 318
-45.2 413
-44.7 418
-5.2 813
-5.1 814
>>
indicating only 1024 and 1369, where there are 4 squares strictly between them, are in the running, and that's when ten is either 795 or 895.
The next square after 795 is 841, and to get there from 795 is to add 46, which contains the forbidden 6.
The next square after 895 is 900, and all that takes is the addition of 5, an allowable digit.
FOUR FIVE TEN FORTUNE
1024 1369 895 1048259
Edited on February 3, 2023, 11:18 am
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Posted by Charlie
on 2023-02-03 11:15:57 |
computer then interpretation
